1503:"measurable"--but depending upon one's field of study, "measurable" means different things. (To someone working with measure in a rather abstract sense, it is probably understood that the σ-algebra of the intermediate space is invariant during the composition; however, to someone working with Lebesgue measure, it is probably understood that the σ-algebra of the intermediate space is different depending on whether it is the domain or the codomain.) I think it is better to be explicit because it seems likely that people with both definitions of "measurable" in mind might read this page, and the answer to the question of composition depends on which definition one uses. Suppressing mention of the underlying σ-algebras is not appropriate here, in my opinion. Oh, and I think it's clear after perusing Billingsley that the σ-algebra on the target space need not be a Borel algebra (although it often is), so I changed this as well. —
84:
74:
53:
2555:
not
Lebesgue measurable are generally considered pathological"). Third, and most importantly, the lede in this particular article probably shouldn't be written in Scientific American style and try to dumb the subject down. In other articles ("Probability Theory" is a good example of that), having a SciA-like intro is great. But measurable function just seems too specialized a notion to be of interest to the non-geeky crowd.
248:. Of course, what you are saying is that in many common situations one uses the Lebesgue σ-algebra for the domain and the Borel σ-algebra for the codomain, and so this consistency is lacking. I think the article needs to explain this clearly, rather than just stating that the composition of two measurable functions may not be a measurable function, since this statement is clearly false when interpreted in the obvious way. --
22:
3264:
2750:. In my thought, in that comment you did not give a reason why we may extend the codomain in the definition of measurable functions from nonnegative reals to an arbitrary measurable space. I am wondering if such extension in the current definition has any meaningful result regarding integration(other than this:
2625:
In my thought, one defines measurable functions to use it to define (Lebesgue) integral. However, to define an integral, doesn't one need a limit process, and thus, a topology? If the codomain of a measurable function is not a topological space but merely a measurable one, is it possible to define a
189:
The current version (Nov4 2006) seems to be contradictory. 1)It says that measurable functions are the morphisms and 2) says composition of measurable funtions needn't be measurable. I guess the measurable functions could be a generating set for the morphisms if (2) is true, in which case (1) should
2657:
In probability theory, you want to assign probabilities to individual events before you start talking about random variables, right? And it's perfectly meaningful to have several RVs attached to the same probability space. To me this means that measure spaces are primary, and measurable functions
2554:
I'm going to re-write the lede because the current one, in my view, doesn't serve its purpose. First, the opening sentence (i.e. "a measurable function is a structure-preserving function between measurable spaces") is misleading. Second, there is some extraneous fluff (e.g. "... functions that are
2194:
As one poster said earlier, this pages suffers from excessive generality. I am studying analysis in graduate school and this page is so convoluted it had me questioning what it meant to be measurable. Could we rewrite the page to focus on
Lebesgue measurability and one or more of the definitions
1502:
what "measurable" means. (This fact should be apparent from the above comments on this page.) I have changed the statement to a more explicit statement which is quite definitely correct, using
Billingsley's notation of "measurable Σ/Τ". I know that no one ever says "measurable Σ/Τ"--one just says
2174:
I usually come to wikipediate maths pages looking for a specific result or definition. Therefore I think it best to include a section that contains a formal definition. Removing the definition from the introductory paragraph would also allow this paragraph to provide a more intuitive idea of the
225:
The key thing to keep in mind is that every subset of a measure-0 set is measurable. That means you can have very pathological sets that are still measurable, just because all their pathology is coded up inside the Cantor set or something. Then you can use a well-behaved function under which the
2775:
On the last question, yes, I did the same thing, and I found out that any of the articles (which I looked at) did not give a source(I mean, any paper, any textbook or at least something outside this WP) of the definition. You know that articles in other language versions of WP tend to be a mere
200:
It's parallel to the composition of cts functions being cts in topology--If f and g are measurable and f is from A to B and g is fr B to C, then g^-1(meas set in C) is measurable in B and f^-1(g^-1(meas set in C)) is measurable in A. Now I know I'm wrong, since I'm close to an idiot in top. and
2793:”, rather than merely a measurable function. The author adds: “We observe that this language is inconsistent with our earlier one concerning measurable functions; …… confusion can always be avoided by use of the proper one of the terms ‘function’ and ‘transformation.’” --
2737:
Yes, if we want to define LI for non-negative measurable functions then the current definition of a measurable function works. However, if the current definition is justified only in that it fits for non-negative ones, then it is in no way generalized than requiring just
194:
on Sep 15, 2006) the statement is apparently backed up by
Mathworld by Todd Rowland and also by Trovatore in answer to a question on the HelpDesk late last month. My obvious reasoning follows, and I'd be an idiot to think I'm right, it's too easy to be missed by you
217:
You'd be right, if the definition of "measurable function" were "the preimage of every measurable set is measurable". But that's too restrictive a definition (doesn't even include all the continuous functions). The standard definition is "the preimage of every
174:
This page has at least a few errors. The composition of two
Lebesgue measurable functions is not neccesarily Lebesgue measurable. Some types of integrals work just fine on non-measurable functions, e.g. a stieltjes integral.(This comment made June 13 2006 by
257:
The article suffers from excessive generality. The most common sense in which the term "measurable function" is used is to mean that the preimage of a Borel set is measurable; that should be the first definition, with generalizations treated later.
715:
As pointed out, the composition of two measurable functions to the reals need not be measurable. First of all, in order to compose two such functions, the second function must be a function from the reals to the reals. But a measurable function
2663:
By the way, since we know that every non-negative measurable function is a pointwise limit of a non-decreasing sequence of simple non-negative functions, what prevents us from using this to define the LI? This is exactly what some authors
1641:
I think it would be illustrative to include in the article a proof sketch of why fg is measurable. I'm trying to learn this stuff and I find it difficult to make the jump from definitions to understanding without anything in between.
1517:
may be more correct, depending on the context. A composition of functions may be measurable or not depending on the domain and the codomain, in the proof presented by
Billingsley I believe that everything is considered measurable.
243:
But in fact the article does define a measurable function to be one in which the pre-image of every measurable set is measurable (and that's the correct definition). So the composition of two measurable functions is measurable,
3090:
1385:) of any two functions that are measurable in the second sense is indeed measurable, and in that sense they form a sensible class of morphisms to set up a category of measurable spaces and measurable functions.
3222:
if you want to add, subtract them, etc. It's a terminology issue more than anything else. I, personally, find it easier to talk about arbitrary measurable functions, but, again, it's a matter of preference.
1864:
To be honest I would call that a "trickological" proof. I don't know what your priority queue is like, but I'd suggest you try to find a proof that's a little more enlightening as to what's going on. --
1853:
2669:
On the last question you asked, I looked at this article ("Measurable function") in several languages, and most, if not all, seem to use (to the best of my understanding) the definition we're using.
2648:
measurability or topology for the definition itself. But how useful this definition is without the two is a question. You at least need the so-called "continuity from below" property to prove the
445:
398:
2962:
2860:
1631:
might not be measurable, but that it might not be a function at all. But for any reasonable way of extending the definitions to partial functions, the result should still be fine, I think. --
1354:
140:
1470:", the preimage of a measurable set is measurable. The latter definition is closed to compositions, while the former is not. Maybe we should do the same - have two interlinked pages.
3272:
2244:
531:
Then what I called statement (1) above needs to be corrected or eliminated. Even if eliminated from this article, a correct statement about the morphisms in the usual category of
507:
476:
312:
1611:
1535:
A useful characterisation of
Lebesgue measurable functions is that f is measurable if and only if mid{-g,f,g} is integrable for all non-negative Lebesgue integrable functions g.
1173:
1141:
1068:
1582:
344:
3180:
3220:
1396:
That second sense needs to be defined explicitly and put in a separate section with the morphism claim, and attention called to the difference in senses(in a good article).
967:
2287:
1234:
856:
2912:
935:
1706:
681:
specialize to the first definition because the first definition only implies that the inverse image of every Borel set is measurable, not every
Lebesgue-measurable set.
3118:
1762:
1734:
1677:
2880:
2472:
is devastating. At any rate, there is an outer measure bearing the name
Lebesgue measure that reduces to (ordinary) Lebesgue measure when restricted to the
3303:
130:
2882:
is not a measure space? Yes, it can be and has (sort of) already been. Use the cumulative distribution function in probability theory as an example. If
2975:
540:
The 'common' definition of measurable
Zundark describes should replace the current one asap. We can use the Folland reference for that, I think.
106:
3298:
226:
preimage of the Cantor set has positive measure, and then the preimage of your pathological set has a chance to "show itself" as it were. --
2144:. (Moreover, there was an inconsistency, since the assumption that the measure is a probability measure was implicit, but not explicit.)
1111:
On the other hand, the proof that the composition of two measurable functions in the other sense is measurable is quite easy: if we have
698:
Confusing matters worse is that usually, when unqualified, a "measurable" function on a space is a function from that space to the reals.
546:
There is some good stuff in this article but overall it is currently not very good. Thanks for the very helpful replies from both of you.
2141:
770:. It relies on the fact that every subset of the reals with positive outer measure contains a nonmeasurable set. Consider the function
3256:
2505:
1982:. This is well defined (and, in particular, unique) for every subset of the codomain of the function. It may be that the preimage of
2159:
97:
58:
1767:
1498:
I think that the statement "the composition of measurable functions is measurable" is misleading because it is not clear
2649:
403:
356:
33:
3280:
2917:
2818:
3240:
2674:
2560:
1239:
2314:\alpah} \in Sigma. Sigma is the sigma-algebra which contains X. from Linear Functional Analysis by B.P. Rynne.
1441:
1431:
1397:
748:
615:
564:
547:
202:
180:
2614:.” Some other textbooks that only deal with real measurable ones require the same one with sets of the form (
2063:
1948:
3268:
1623:
Well, those wouldn't be "real-valued"; they'd be "extended-real-valued", and the problem would be not that
3276:
2798:
2634:
2319:
3236:
3235:
3. What makes you think everyone blindly translates math articles from English into their own language?
2670:
2556:
2298:
2198:
2180:
2155:
1414:
1357:
163:
39:
1489:
1440:
I removed the morphisms statement until it can be clarified and corrected, but forgot the edit summary.
543:
A simple example f(x) if there is one, with f measurable and f(f(x)) not measurable, would be helpful.
481:
450:
281:
83:
3275:
until a consensus is reached, and readers of this page are welcome to contribute to the discussion.
2794:
2697:
2630:
2147:
2104:
isn't in the range) then there is no contribution to the preimage set, and if multiple points map to
1587:
1146:
1114:
972:
1561:
603:,Σ) to the reals is defined to be measurable when the inverse image of every open set is measurable.
176:
21:
2442:
2176:
1514:
1504:
325:
3139:
191:
105:
on Knowledge. If you would like to participate, please visit the project page, where you can join
2515:
2059:
1944:
201:
analysis, but it will improve the article to have an explanation and references for this. Thanks,
89:
3185:
2485:-measurable and thereby put Lebesgue measurable function in a context. A possible reference is
940:
73:
52:
2540:
2502:
2253:
1194:
825:
2885:
914:
2294:
2151:
1682:
563:
I'll put in an expert needed template to get an expert and give innocent readers a heads up.
3103:
2528:
2498:
2490:
1739:
1711:
805:
222:
set is measurable" (and thus automatically the preimage of every Borel set is measurable).
1660:
736:
of a Lebesgue-measurable set Lebesgue-measurable. I can find you an example if you want.
2315:
1865:
1856:
1643:
1632:
1614:
1539:
522:
259:
227:
2865:
3292:
2392:
2340:
2120:
1997:
1471:
747:
If you can find one or tell me a promising place to look for one that would be great.
660:,Τ) is defined to be measurable if the inverse image of every set in Τ is a set in Σ.
532:
3120:
in the general case. Now, whether this definition is useful is a different question.
2536:
249:
3133:
2. This might be too trivial, but you can always talk about measurable function
3085:{\displaystyle \int _{f(X)}f\,d\mu _{Y}=\sum _{i=1}^{n}c_{i}\mu _{Y}(\{c_{i}\})}
102:
3284:
3244:
2802:
2678:
2638:
2564:
2544:
2497:. Classics in Mathematics (1st ed reprint ed.). Berlin, Heidelberg, New York:
2323:
2302:
2184:
2163:
2125:
2080:
For a countable set, that is exactly the right idea, with the caveats that if
2067:
2002:
1952:
1868:
1859:
1646:
1635:
1617:
1542:
1522:
1519:
1507:
1492:
1485:
1474:
1444:
1434:
1417:
1400:
1360:
813:
751:
618:
567:
550:
525:
262:
252:
230:
205:
183:
79:
2313:
R is said to be measurable if, for every \alpha \in R, {x \in X : f(x) : -->
3273:
Knowledge:Redirects for discussion/Log/2021 January 11#Bimeasurable function
2411:
1555:
The sum and product of two real-valued measurable functions are measurable.
2789:, a function satisfying the current WP definition is called a “measurable
513:
understood as the σ-algebra on the range space unless otherwise specified.
2629:
Does someone know the source of the current version of the definition? --
2479:
Provided my observations make sense, perhaps the article could introduce
2116:
2012:
1993:
3263:
190:
be modified. I don't see why (2) should true, but (Statement (2) was by
1451:
2018:
so, in case it's not clear from my original question, I presumed that
614:
The very last word in the preceding sentence is . I'm sure you meant .
2696:
By the way, don't you mean by “The WP definition of LI” this section
2464:
is Borel regular. I'm not too sure whether the distinction between
1986:
is the empty set, for example, if no point in the domain maps into
162:
Borel function - that para really needs turning round to be clear.
1484:
3rd ed. , Patrick Billingsley, Wiley, 1995. ISBN:0-471-00710-2.
2582:, the definition of a measurable function goes like “a function
1480:
Please see the following and perhaps this will help: page 182
170:
Composition of measurable functions not necessarily measurable
15:
1558:
What about when we have functions f,g and f+g is of the form
1466:(hence, Borel) set is measurable, whereas with a measurable "
2626:
useful integral? Then why one defines measurable functions?
732:
of a Borel set Lebesgue-measurable; it doesn't usually have
488:
457:
423:
412:
376:
365:
331:
296:
246:
as long as you are consistent about the σ-algebra being used
582:
Okay, but it is otherwise specified later. Here's the rub.
314:
is a measurable space, a real- or complex-valued function
1848:{\displaystyle fg={\frac {1}{4}}((f+g)^{2}+(f-g)^{2})\,}
2644:
The WP definition of LI is set-theoretic, so you don't
2441:, then the above reduces to the content of the section
519:
Real Analysis: Modern Techniques and Their Applications
2921:
2822:
2410:-measurable if and only if the inverse image of every
2379:-measurable if the inverse image of every open set is
3267:
A discussion is taking place to address the redirect
3188:
3142:
3106:
2978:
2920:
2888:
2868:
2821:
2728:
measurability or topology for the definition itself”.
2256:
2201:
2113:
then they all get put into the preimage set. — Carl
1770:
1742:
1714:
1685:
1663:
1590:
1564:
1242:
1197:
1149:
1117:
1076:
is a measurable set that is not Borel, because since
975:
943:
917:
828:
766:
The example I know is from the exercises in Royden's
484:
453:
406:
359:
328:
284:
101:, a collaborative effort to improve the coverage of
2454:-measurable function makes much sense, at least if
2293:], then move on to definitions in full generality?
1654:
Here it is if someone wants to fill in the details:
1548:
Sum and product of real-valued measurable functions
440:{\displaystyle ({\mathcal {M}},{\mathcal {B}}_{C})}
393:{\displaystyle ({\mathcal {M}},{\mathcal {B}}_{R})}
3214:
3174:
3112:
3084:
2956:
2906:
2874:
2854:
2281:
2238:
1847:
1756:
1728:
1700:
1671:
1605:
1576:
1348:
1228:
1167:
1135:
1062:
961:
929:
850:
501:
470:
439:
392:
338:
306:
2957:{\displaystyle \textstyle \int _{X}f\,d\mu _{X},}
2914:is a simple function, then, rather than defining
2724:. So I don't understand this comment: “you don't
1462:. With a measurable function, the preimage of an
3107:
2855:{\displaystyle \textstyle \int _{X}f\,d\mu _{X}}
2698:Lebesgue_integration#Towards_a_formal_definition
2622:, which form subbase for the standard topology.
2815:1. The way I understood your question was: can
1538:Why is it useful? What does mid{-g,f,g} mean?
1450:Well, well. The SpringerLink Maths dictionary (
2140:because it duplicates some of the material in
1349:{\displaystyle (h\circ g)^{-1}=g^{-1}]=g^{-1}}
8:
3076:
3063:
2136:Deleted Section "Stationary transformations"
1513:I think that the previous statement made by
2652:. The MCT is then used to prove linearity.
1708:is measurable for integer powers k. Since
2011:Ah, I was unacquainted with the notion of
47:
3206:
3193:
3187:
3163:
3150:
3141:
3105:
3070:
3054:
3044:
3034:
3023:
3010:
2983:
2977:
2944:
2926:
2919:
2887:
2867:
2845:
2827:
2820:
2339:-measurable. At least I have seen it for
2261:
2255:
2206:
2200:
1834:
1809:
1780:
1769:
1741:
1713:
1690:
1684:
1662:
1589:
1563:
1454:) makes the useful distinction between a
1328:
1300:
1284:
1259:
1241:
1208:
1196:
1148:
1116:
1036:
996:
980:
974:
942:
916:
839:
827:
493:
487:
486:
483:
462:
456:
455:
452:
428:
422:
421:
411:
410:
405:
381:
375:
374:
364:
363:
358:
330:
329:
327:
295:
294:
283:
2578:(3e) or Johnsonbaugh and Pfaffenberger,
2170:Separate section with formal definition?
1935:was injective, so how can we talk about
1183:for each map, I mean), then if a subset
3001:
2935:
2836:
1919:. Two questions... first, nobody said
1843:
1752:
1724:
1696:
1667:
49:
19:
2700:? This definition becomes nonsense if
2524:
2513:
2333:There is a notion of a function being
892:is a subset of a set of measure zero (
2329:Measurable and μ-measurable functions
937:is not, because the inverse image of
7:
2776:translation of this English version.
2580:Foundations of Mathematical Analysis
1452:http://eom.springer.de/M/m063210.htm
903:We know the characteristic function
858:is continuous, and hence measurable.
95:This article is within the scope of
2574:In textbooks such as Walter Rudin,
2359:be a topological space. A function
2239:{\displaystyle f^{-1}((x,\infty ))}
2142:measure-preserving dynamical system
1943:? Am I missing something obvious?—
1088:was (and it's not even measurable).
633:Right. Whoops. I've changed it now.
535:should be inserted in that article.
38:It is of interest to the following
3304:High-priority mathematics articles
3277:𝟙𝟤𝟯𝟺𝐪𝑤𝒆𝓇𝟷𝟮𝟥𝟜𝓺𝔴𝕖𝖗𝟰
3255:"Bimeasurable function" listed at
2766:non-negative Σ-measurable) or not.
2227:
1600:
1594:
1571:
1565:
1468:mapping between two measure spaces
1017:
953:
502:{\displaystyle {\mathcal {B}}_{C}}
471:{\displaystyle {\mathcal {B}}_{R}}
307:{\displaystyle (X,{\mathcal {M}})}
14:
2785:For instance, in Paul R. Halmos,
2289:is measurable for all Borel sets
1931:a preimage. Second, nobody said
884:that is nonmeasurable, but since
652:A function from a measure space (
599:A function from a measure space (
115:Knowledge:WikiProject Mathematics
3262:
3182:instead of measurable functions
2618:, ∞] in place of every open set
2036:, ...}) was being used to mean {
1606:{\displaystyle -\infty +\infty }
1175:both measurable (using the same
1168:{\displaystyle h:Y\rightarrow Z}
1136:{\displaystyle g:X\rightarrow Y}
1063:{\displaystyle g^{-1}]=g^{-1}=B}
274:Here's what Folland has to say:
118:Template:WikiProject Mathematics
82:
72:
51:
20:
3271:. The discussion will occur at
1923:was surjective, so in general,
1899:provided that for every subset
1764:is measurable. Then note that
1577:{\displaystyle \infty -\infty }
135:This article has been rated as
3169:
3143:
3079:
3060:
2993:
2987:
2898:
2658:are secondary, not vice versa.
2349:be a space with outer measure
2276:
2270:
2233:
2230:
2218:
2215:
1840:
1831:
1818:
1806:
1793:
1790:
1343:
1337:
1318:
1315:
1309:
1293:
1274:
1268:
1256:
1243:
1223:
1217:
1159:
1127:
1051:
1045:
1026:
1023:
1020:
1008:
1005:
989:
956:
944:
656:,Σ) to another measure space (
434:
407:
387:
360:
339:{\displaystyle {\mathcal {M}}}
301:
285:
1:
3175:{\displaystyle (f^{+},f^{-})}
2803:05:36, 27 December 2017 (UTC)
2679:21:37, 20 December 2017 (UTC)
2639:22:24, 19 December 2017 (UTC)
2476:-algebra of measurable sets.
2423:Now, as far as I can see, if
2126:14:34, 28 February 2008 (UTC)
2068:04:23, 28 February 2008 (UTC)
2003:01:56, 28 February 2008 (UTC)
1953:01:26, 28 February 2008 (UTC)
109:and see a list of open tasks.
3299:C-Class mathematics articles
3285:13:09, 11 January 2021 (UTC)
2758:non-negative Τ-measurable ⇒
2687:Thank you for kind response.
2650:monotone convergence theorem
2324:00:20, 8 February 2012 (UTC)
1869:20:49, 15 October 2007 (UTC)
1860:01:17, 15 October 2007 (UTC)
1647:23:21, 14 October 2007 (UTC)
1636:19:10, 14 October 2007 (UTC)
1618:04:24, 14 October 2007 (UTC)
1543:04:02, 14 October 2007 (UTC)
1493:04:33, 6 February 2007 (UTC)
1475:23:13, 17 January 2007 (UTC)
1445:00:53, 5 November 2006 (UTC)
1435:02:15, 5 November 2006 (UTC)
1418:00:51, 5 November 2006 (UTC)
1401:02:15, 5 November 2006 (UTC)
1361:07:24, 5 November 2006 (UTC)
752:02:15, 5 November 2006 (UTC)
728:in the usual sense only has
619:02:15, 5 November 2006 (UTC)
568:00:39, 5 November 2006 (UTC)
551:00:21, 5 November 2006 (UTC)
526:20:35, 4 November 2006 (UTC)
263:20:09, 4 November 2006 (UTC)
253:19:36, 4 November 2006 (UTC)
231:19:00, 4 November 2006 (UTC)
206:09:57, 4 November 2006 (UTC)
184:09:46, 4 November 2006 (UTC)
3245:03:04, 2 January 2018 (UTC)
3215:{\displaystyle f^{+}-f^{-}}
2746:, ∞] is measurable for all
2460:is a topological space and
1488:21:42, 2 August 2007 (UTC)
962:{\displaystyle (0,\infty )}
869:has measure 1, even though
3320:
2570:Is the definition correct?
2565:21:53, 3 August 2017 (UTC)
2400:Then it is a theorem that
1523:21:42, 2 August 2007 (UTC)
1381:However, the composition (
2576:Real and Complex Analysis
2545:10:01, 11 July 2016 (UTC)
2303:21:58, 20 July 2010 (UTC)
2282:{\displaystyle f^{-1}(B)}
2185:22:44, 12 July 2010 (UTC)
2164:16:42, 8 April 2008 (UTC)
2091:) doesn't exist for some
1508:16:13, 30 July 2007 (UTC)
1229:{\displaystyle E:=h^{-1}}
851:{\displaystyle g:=f^{-1}}
134:
67:
46:
3257:Redirects for discussion
2907:{\displaystyle f:X\to Y}
2586:from a measurable space
2495:Geometric Measure Theory
1970:in the domain such that
930:{\displaystyle h\circ g}
166:12:48, 6 Sep 2004 (UTC)
141:project's priority scale
2590:to a topological space
1701:{\displaystyle f^{k}\,}
1482:Probability and Measure
1191:is measurable, we know
1070:which isn't measurable.
822:is a homeomorphism, so
677:This second definition
98:WikiProject Mathematics
3216:
3176:
3114:
3086:
3039:
2958:
2908:
2876:
2856:
2283:
2246:is measurable for all
2240:
1958:The preimage of a set
1883:The article says that
1849:
1758:
1730:
1702:
1673:
1607:
1578:
1350:
1236:is measurable, and so
1230:
1169:
1137:
1064:
963:
931:
852:
503:
472:
441:
394:
340:
308:
28:This article is rated
3269:Bimeasurable function
3217:
3177:
3115:
3113:{\displaystyle \sup }
3087:
3019:
2959:
2909:
2877:
2857:
2308:Different Definition?
2284:
2241:
1966:is the set of points
1879:Glitch in definition?
1850:
1759:
1757:{\displaystyle f+g\,}
1731:
1729:{\displaystyle f,g\,}
1703:
1674:
1608:
1579:
1351:
1231:
1170:
1138:
1065:
964:
932:
853:
504:
473:
442:
395:
341:
309:
3186:
3140:
3104:
2976:
2918:
2886:
2866:
2819:
2312:A function f:X-: -->
2254:
2199:
1768:
1740:
1712:
1683:
1661:
1588:
1562:
1528:What is mid{-g,f,g}?
1430:Yes it does, thanks.
1240:
1195:
1179:-algebra of sets on
1147:
1115:
1080:is a homeomorphism,
973:
941:
915:
826:
482:
451:
404:
357:
326:
282:
121:mathematics articles
2606:for every open set
2602:) is measurable in
2550:Re-writing the lede
1672:{\displaystyle f\,}
1456:measurable function
911:is measurable, but
517:Folland, Gerald B.
3212:
3172:
3110:
3082:
3002:
2954:
2953:
2936:
2904:
2872:
2852:
2851:
2837:
2754:: Σ/Τ-measurable,
2708:in the sense that
2395:, don't remember.)
2391:is required to be
2279:
2236:
1845:
1844:
1754:
1753:
1726:
1725:
1698:
1697:
1669:
1668:
1603:
1574:
1460:measurable mapping
1413:Hope this helps. —
1346:
1226:
1165:
1133:
1084:would be Borel if
1060:
969:under that map is
959:
927:
876:There is a subset
848:
499:
468:
437:
390:
336:
304:
90:Mathematics portal
34:content assessment
2875:{\displaystyle X}
2399:
2166:
2150:comment added by
2124:
2001:
1962:under a function
1788:
1552:From the article:
1532:From the article:
521:, 1984, p. 43. --
155:
154:
151:
150:
147:
146:
3311:
3266:
3237:StrokeOfMidnight
3221:
3219:
3218:
3213:
3211:
3210:
3198:
3197:
3181:
3179:
3178:
3173:
3168:
3167:
3155:
3154:
3119:
3117:
3116:
3111:
3091:
3089:
3088:
3083:
3075:
3074:
3059:
3058:
3049:
3048:
3038:
3033:
3015:
3014:
2997:
2996:
2963:
2961:
2960:
2955:
2949:
2948:
2931:
2930:
2913:
2911:
2910:
2905:
2881:
2879:
2878:
2873:
2862:be defined when
2861:
2859:
2858:
2853:
2850:
2849:
2832:
2831:
2671:StrokeOfMidnight
2557:StrokeOfMidnight
2532:
2526:
2521:
2519:
2511:
2484:
2475:
2463:
2459:
2453:
2440:
2433:
2419:
2409:
2403:
2397:
2390:
2384:
2378:
2372:
2358:
2352:
2348:
2338:
2288:
2286:
2285:
2280:
2269:
2268:
2245:
2243:
2242:
2237:
2214:
2213:
2145:
2114:
1991:
1911:the preimage of
1854:
1852:
1851:
1846:
1839:
1838:
1814:
1813:
1789:
1781:
1763:
1761:
1760:
1755:
1736:are measurable,
1735:
1733:
1732:
1727:
1707:
1705:
1704:
1699:
1695:
1694:
1678:
1676:
1675:
1670:
1612:
1610:
1609:
1604:
1583:
1581:
1580:
1575:
1415:vivacissamamente
1358:vivacissamamente
1356:is measurable. —
1355:
1353:
1352:
1347:
1336:
1335:
1308:
1307:
1292:
1291:
1267:
1266:
1235:
1233:
1232:
1227:
1216:
1215:
1174:
1172:
1171:
1166:
1142:
1140:
1139:
1134:
1069:
1067:
1066:
1061:
1044:
1043:
1004:
1003:
988:
987:
968:
966:
965:
960:
936:
934:
933:
928:
857:
855:
854:
849:
847:
846:
508:
506:
505:
500:
498:
497:
492:
491:
477:
475:
474:
469:
467:
466:
461:
460:
446:
444:
443:
438:
433:
432:
427:
426:
416:
415:
399:
397:
396:
391:
386:
385:
380:
379:
369:
368:
345:
343:
342:
337:
335:
334:
313:
311:
310:
305:
300:
299:
164:Charles Matthews
123:
122:
119:
116:
113:
92:
87:
86:
76:
69:
68:
63:
55:
48:
31:
25:
24:
16:
3319:
3318:
3314:
3313:
3312:
3310:
3309:
3308:
3289:
3288:
3260:
3202:
3189:
3184:
3183:
3159:
3146:
3138:
3137:
3102:
3101:
3066:
3050:
3040:
3006:
2979:
2974:
2973:
2940:
2922:
2916:
2915:
2884:
2883:
2864:
2863:
2841:
2823:
2817:
2816:
2572:
2552:
2522:
2512:
2508:
2499:Springer Verlag
2489:
2480:
2473:
2461:
2455:
2449:
2435:
2424:
2415:
2405:
2401:
2388:
2380:
2374:
2360:
2354:
2350:
2344:
2334:
2331:
2310:
2257:
2252:
2251:
2202:
2197:
2196:
2195:for that [e.g.
2192:
2172:
2138:
2112:
2103:
2090:
2057:
2046:
2035:
2028:
1881:
1830:
1805:
1766:
1765:
1738:
1737:
1710:
1709:
1686:
1681:
1680:
1679:is measurable,
1659:
1658:
1586:
1585:
1560:
1559:
1550:
1530:
1324:
1296:
1280:
1255:
1238:
1237:
1204:
1193:
1192:
1145:
1144:
1113:
1112:
1032:
992:
976:
971:
970:
939:
938:
913:
912:
835:
824:
823:
806:Cantor function
803:
792:
774:: → given by
485:
480:
479:
454:
449:
448:
420:
402:
401:
373:
355:
354:
324:
323:
322:will be called
280:
279:
172:
160:
120:
117:
114:
111:
110:
88:
81:
61:
32:on Knowledge's
29:
12:
11:
5:
3317:
3315:
3307:
3306:
3301:
3291:
3290:
3259:
3253:
3252:
3251:
3250:
3249:
3248:
3247:
3228:
3227:
3226:
3225:
3224:
3223:
3209:
3205:
3201:
3196:
3192:
3171:
3166:
3162:
3158:
3153:
3149:
3145:
3126:
3125:
3124:
3123:
3122:
3121:
3109:
3093:
3092:
3081:
3078:
3073:
3069:
3065:
3062:
3057:
3053:
3047:
3043:
3037:
3032:
3029:
3026:
3022:
3018:
3013:
3009:
3005:
3000:
2995:
2992:
2989:
2986:
2982:
2970:
2969:
2968:
2967:
2966:
2965:
2952:
2947:
2943:
2939:
2934:
2929:
2925:
2903:
2900:
2897:
2894:
2891:
2871:
2848:
2844:
2840:
2835:
2830:
2826:
2808:
2807:
2806:
2805:
2791:transformation
2787:Measure Theory
2780:
2779:
2778:
2777:
2770:
2769:
2768:
2767:
2732:
2731:
2730:
2729:
2691:
2690:
2689:
2688:
2682:
2681:
2666:
2665:
2660:
2659:
2654:
2653:
2571:
2568:
2551:
2548:
2534:
2533:
2507:978-3540606567
2506:
2448:The notion of
2341:outer measures
2330:
2327:
2309:
2306:
2278:
2275:
2272:
2267:
2264:
2260:
2235:
2232:
2229:
2226:
2223:
2220:
2217:
2212:
2209:
2205:
2191:
2188:
2171:
2168:
2137:
2134:
2133:
2132:
2131:
2130:
2129:
2128:
2108:
2099:
2088:
2073:
2072:
2071:
2070:
2055:
2044:
2033:
2026:
2006:
2005:
1880:
1877:
1876:
1875:
1874:
1873:
1872:
1871:
1842:
1837:
1833:
1829:
1826:
1823:
1820:
1817:
1812:
1808:
1804:
1801:
1798:
1795:
1792:
1787:
1784:
1779:
1776:
1773:
1751:
1748:
1745:
1723:
1720:
1717:
1693:
1689:
1666:
1655:
1639:
1638:
1602:
1599:
1596:
1593:
1573:
1570:
1567:
1557:
1553:
1549:
1546:
1537:
1533:
1529:
1526:
1511:
1510:
1478:
1477:
1438:
1437:
1427:
1426:
1425:
1424:
1423:
1422:
1421:
1420:
1404:
1403:
1393:
1392:
1391:
1390:
1389:
1388:
1387:
1386:
1372:
1371:
1370:
1369:
1368:
1367:
1366:
1365:
1364:
1363:
1345:
1342:
1339:
1334:
1331:
1327:
1323:
1320:
1317:
1314:
1311:
1306:
1303:
1299:
1295:
1290:
1287:
1283:
1279:
1276:
1273:
1270:
1265:
1262:
1258:
1254:
1251:
1248:
1245:
1225:
1222:
1219:
1214:
1211:
1207:
1203:
1200:
1164:
1161:
1158:
1155:
1152:
1132:
1129:
1126:
1123:
1120:
1100:
1099:
1098:
1097:
1096:
1095:
1094:
1093:
1092:
1091:
1090:
1089:
1071:
1059:
1056:
1053:
1050:
1047:
1042:
1039:
1035:
1031:
1028:
1025:
1022:
1019:
1016:
1013:
1010:
1007:
1002:
999:
995:
991:
986:
983:
979:
958:
955:
952:
949:
946:
926:
923:
920:
901:
900:is measurable.
874:
873:has measure 0.
859:
845:
842:
838:
834:
831:
801:
790:
755:
754:
744:
743:
742:
741:
740:
739:
738:
737:
706:
705:
704:
703:
702:
701:
700:
699:
689:
688:
687:
686:
685:
684:
683:
682:
668:
667:
666:
665:
664:
663:
662:
661:
643:
642:
641:
640:
639:
638:
637:
636:
635:
634:
622:
621:
611:
610:
609:
608:
607:
606:
605:
604:
590:
589:
588:
587:
586:
585:
584:
583:
573:
572:
571:
570:
558:
557:
556:
555:
554:
553:
537:
536:
533:measure spaces
515:
514:
496:
490:
465:
459:
436:
431:
425:
419:
414:
409:
389:
384:
378:
372:
367:
362:
333:
303:
298:
293:
290:
287:
272:
271:
270:
269:
268:
267:
266:
265:
236:
235:
234:
233:
223:
215:
209:
208:
197:
196:
177:66.171.165.218
171:
168:
159:
158:Borel function
156:
153:
152:
149:
148:
145:
144:
133:
127:
126:
124:
107:the discussion
94:
93:
77:
65:
64:
56:
44:
43:
37:
26:
13:
10:
9:
6:
4:
3:
2:
3316:
3305:
3302:
3300:
3297:
3296:
3294:
3287:
3286:
3282:
3278:
3274:
3270:
3265:
3258:
3254:
3246:
3242:
3238:
3234:
3233:
3232:
3231:
3230:
3229:
3207:
3203:
3199:
3194:
3190:
3164:
3160:
3156:
3151:
3147:
3136:
3132:
3131:
3130:
3129:
3128:
3127:
3099:
3098:
3097:
3096:
3095:
3094:
3071:
3067:
3055:
3051:
3045:
3041:
3035:
3030:
3027:
3024:
3020:
3016:
3011:
3007:
3003:
2998:
2990:
2984:
2980:
2972:
2971:
2950:
2945:
2941:
2937:
2932:
2927:
2923:
2901:
2895:
2892:
2889:
2869:
2846:
2842:
2838:
2833:
2828:
2824:
2814:
2813:
2812:
2811:
2810:
2809:
2804:
2800:
2796:
2792:
2788:
2784:
2783:
2782:
2781:
2774:
2773:
2772:
2771:
2765:
2761:
2757:
2753:
2749:
2745:
2741:
2736:
2735:
2734:
2733:
2727:
2723:
2719:
2715:
2711:
2707:
2703:
2699:
2695:
2694:
2693:
2692:
2686:
2685:
2684:
2683:
2680:
2676:
2672:
2668:
2667:
2662:
2661:
2656:
2655:
2651:
2647:
2643:
2642:
2641:
2640:
2636:
2632:
2627:
2623:
2621:
2617:
2613:
2609:
2605:
2601:
2597:
2593:
2589:
2585:
2581:
2577:
2569:
2567:
2566:
2562:
2558:
2549:
2547:
2546:
2542:
2538:
2530:
2517:
2509:
2504:
2500:
2496:
2492:
2488:
2487:
2486:
2483:
2477:
2471:
2470:outer measure
2467:
2458:
2452:
2446:
2444:
2438:
2431:
2427:
2421:
2420:-measurable.
2418:
2413:
2408:
2396:
2394:
2393:Borel regular
2385:-measurable.
2383:
2377:
2371:
2367:
2363:
2357:
2347:
2342:
2337:
2328:
2326:
2325:
2321:
2317:
2307:
2305:
2304:
2300:
2296:
2292:
2273:
2265:
2262:
2258:
2249:
2224:
2221:
2210:
2207:
2203:
2189:
2187:
2186:
2182:
2178:
2169:
2167:
2165:
2161:
2157:
2153:
2149:
2143:
2135:
2127:
2122:
2118:
2111:
2107:
2102:
2098:
2094:
2087:
2083:
2079:
2078:
2077:
2076:
2075:
2074:
2069:
2065:
2061:
2060:PaulTanenbaum
2054:
2050:
2043:
2039:
2032:
2025:
2021:
2017:
2014:
2010:
2009:
2008:
2007:
2004:
1999:
1995:
1989:
1985:
1981:
1977:
1973:
1969:
1965:
1961:
1957:
1956:
1955:
1954:
1950:
1946:
1945:PaulTanenbaum
1942:
1938:
1934:
1930:
1927:doesn't even
1926:
1922:
1918:
1914:
1910:
1906:
1902:
1898:
1894:
1890:
1886:
1878:
1870:
1867:
1863:
1862:
1861:
1858:
1835:
1827:
1824:
1821:
1815:
1810:
1802:
1799:
1796:
1785:
1782:
1777:
1774:
1771:
1749:
1746:
1743:
1721:
1718:
1715:
1691:
1687:
1664:
1656:
1653:
1652:
1651:
1650:
1649:
1648:
1645:
1637:
1634:
1630:
1626:
1622:
1621:
1620:
1619:
1616:
1597:
1591:
1568:
1556:
1547:
1545:
1544:
1541:
1536:
1527:
1525:
1524:
1521:
1516:
1509:
1506:
1501:
1497:
1496:
1495:
1494:
1491:
1487:
1483:
1476:
1473:
1469:
1465:
1461:
1457:
1453:
1449:
1448:
1447:
1446:
1443:
1436:
1433:
1429:
1428:
1419:
1416:
1412:
1411:
1410:
1409:
1408:
1407:
1406:
1405:
1402:
1399:
1395:
1394:
1384:
1383:where defined
1380:
1379:
1378:
1377:
1376:
1375:
1374:
1373:
1362:
1359:
1340:
1332:
1329:
1325:
1321:
1312:
1304:
1301:
1297:
1288:
1285:
1281:
1277:
1271:
1263:
1260:
1252:
1249:
1246:
1220:
1212:
1209:
1205:
1201:
1198:
1190:
1186:
1182:
1178:
1162:
1156:
1153:
1150:
1130:
1124:
1121:
1118:
1110:
1109:
1108:
1107:
1106:
1105:
1104:
1103:
1102:
1101:
1087:
1083:
1079:
1075:
1072:
1057:
1054:
1048:
1040:
1037:
1033:
1029:
1014:
1011:
1000:
997:
993:
984:
981:
977:
950:
947:
924:
921:
918:
910:
906:
902:
899:
895:
891:
887:
883:
879:
875:
872:
868:
864:
860:
843:
840:
836:
832:
829:
821:
818:
817:
815:
811:
807:
800:
796:
789:
785:
781:
777:
773:
769:
768:Real Analysis
765:
764:
763:
762:
761:
760:
759:
758:
757:
756:
753:
750:
746:
745:
735:
731:
727:
723:
719:
714:
713:
712:
711:
710:
709:
708:
707:
697:
696:
695:
694:
693:
692:
691:
690:
680:
676:
675:
674:
673:
672:
671:
670:
669:
659:
655:
651:
650:
649:
648:
647:
646:
645:
644:
632:
631:
630:
629:
628:
627:
626:
625:
624:
623:
620:
617:
613:
612:
602:
598:
597:
596:
595:
594:
593:
592:
591:
581:
580:
579:
578:
577:
576:
575:
574:
569:
566:
562:
561:
560:
559:
552:
549:
545:
544:
542:
541:
539:
538:
534:
530:
529:
528:
527:
524:
520:
512:
494:
463:
429:
417:
382:
370:
352:
348:
321:
317:
291:
288:
277:
276:
275:
264:
261:
256:
255:
254:
251:
247:
242:
241:
240:
239:
238:
237:
232:
229:
224:
221:
216:
213:
212:
211:
210:
207:
204:
199:
198:
193:
188:
187:
186:
185:
182:
178:
169:
167:
165:
157:
142:
138:
137:High-priority
132:
129:
128:
125:
108:
104:
100:
99:
91:
85:
80:
78:
75:
71:
70:
66:
62:High‑priority
60:
57:
54:
50:
45:
41:
35:
27:
23:
18:
17:
3261:
3134:
2790:
2786:
2763:
2759:
2755:
2751:
2747:
2743:
2739:
2725:
2721:
2717:
2713:
2709:
2705:
2701:
2645:
2628:
2624:
2619:
2615:
2611:
2607:
2603:
2599:
2595:
2591:
2587:
2583:
2579:
2575:
2573:
2553:
2535:
2494:
2481:
2478:
2469:
2465:
2456:
2450:
2447:
2436:
2429:
2425:
2422:
2416:
2406:
2398:No it isn't.
2386:
2381:
2375:
2369:
2365:
2361:
2355:
2345:
2335:
2332:
2311:
2290:
2247:
2193:
2173:
2139:
2109:
2105:
2100:
2096:
2092:
2085:
2081:
2052:
2048:
2041:
2037:
2030:
2023:
2019:
2015:
1987:
1983:
1979:
1975:
1971:
1967:
1963:
1959:
1940:
1939:preimage of
1936:
1932:
1928:
1924:
1920:
1916:
1912:
1908:
1904:
1900:
1896:
1892:
1888:
1884:
1882:
1640:
1628:
1624:
1554:
1551:
1534:
1531:
1512:
1499:
1490:68.35.224.53
1481:
1479:
1467:
1463:
1459:
1455:
1439:
1382:
1188:
1184:
1180:
1176:
1085:
1081:
1077:
1073:
908:
904:
897:
893:
889:
885:
881:
877:
870:
866:
862:
819:
809:
798:
794:
787:
783:
779:
775:
771:
767:
733:
729:
725:
721:
717:
678:
657:
653:
600:
518:
516:
510:
447:measurable.
350:
346:
319:
315:
273:
245:
219:
192:128.30.51.97
173:
161:
136:
96:
40:WikiProjects
2491:Federer, H.
2439:= Lebesquge
2295:Ckhenderson
2152:OdedSchramm
2146:—Preceding
353:, if it is
347:-measurable
112:Mathematics
103:mathematics
59:Mathematics
3293:Categories
2795:Inactivate
2718:measurable
2706:measurable
2631:Inactivate
2594:such that
1897:measurable
861:The image
814:Cantor set
351:measurable
349:, or just
2964:we define
2525:|ref=harv
2516:cite book
2493:(1996) .
2412:Borel set
2387:(Perhaps
2316:Brydustin
2175:concept.
2095:(because
2058:), ...}.—
1990:. — Carl
1866:Trovatore
1857:Timhoooey
1644:Timhoooey
1633:Trovatore
1615:Timhoooey
1540:Timhoooey
797:), where
523:Trovatore
260:Trovatore
228:Trovatore
3281:𝗍𝗮𝘭𝙠
3100:and use
2720:for all
2716:, ∞] is
2523:Invalid
2353:and let
2190:Rewrite?
2177:BenWhale
2160:contribs
2148:unsigned
2016:of a set
2013:preimage
1978:) is in
1515:Babcockd
1505:Babcockd
1500:a priori
1472:mousomer
214:Hi Rich,
2704:is not
2537:YohanN7
2466:measure
2434:and if
812:be the
804:is the
679:doesn't
250:Zundark
139:on the
30:C-class
2443:Caveat
2343:. Let
1915:is in
1657:Since
1458:and a
808:. Let
511:always
36:scale.
3135:pairs
1520:Jka02
1486:Jka02
720:from
195:guys:
3241:talk
2799:talk
2726:need
2675:talk
2646:need
2635:talk
2561:talk
2541:talk
2529:help
2503:ISBN
2468:and
2320:talk
2299:talk
2181:talk
2156:talk
2121:talk
2064:talk
1998:talk
1949:talk
1929:have
1464:open
1442:Rich
1432:Rich
1398:Rich
1143:and
782:) =
749:Rich
616:Rich
565:Rich
548:Rich
220:open
203:Rich
181:Rich
131:High
3108:sup
2664:do.
2610:in
2432:= ℝ
2414:is
2404:is
2373:is
2250:or
2117:CBM
2047:),
1994:CBM
1937:the
1907:in
1903:of
1895:is
1613:? -
1584:or
1187:of
907:of
896:),
888::=
880:of
865::=
724:to
509:is
478:or
400:or
318:on
278:If
3295::
3283:)
3243:)
3208:−
3200:−
3165:−
3052:μ
3021:∑
3008:μ
2981:∫
2942:μ
2924:∫
2899:→
2843:μ
2825:∫
2801:)
2677:)
2637:)
2563:)
2543:)
2520::
2518:}}
2514:{{
2501:.
2445:.
2428:=
2368:→
2322:)
2301:)
2263:−
2228:∞
2208:−
2183:)
2162:)
2158:•
2119:·
2066:)
2029:,
2022:({
1996:·
1951:)
1891:→
1887::
1855:.
1825:−
1601:∞
1595:∞
1592:−
1572:∞
1569:−
1566:∞
1330:−
1302:−
1286:−
1261:−
1250:∘
1210:−
1202::=
1160:→
1128:→
1038:−
1018:∞
998:−
982:−
954:∞
922:∘
841:−
833::=
816:.
786:+
258:--
3279:(
3239:(
3204:f
3195:+
3191:f
3170:)
3161:f
3157:,
3152:+
3148:f
3144:(
3080:)
3077:}
3072:i
3068:c
3064:{
3061:(
3056:Y
3046:i
3042:c
3036:n
3031:1
3028:=
3025:i
3017:=
3012:Y
3004:d
2999:f
2994:)
2991:X
2988:(
2985:f
2951:,
2946:X
2938:d
2933:f
2928:X
2902:Y
2896:X
2893::
2890:f
2870:X
2847:X
2839:d
2834:f
2829:X
2797:(
2764:f
2762:∘
2760:g
2756:g
2752:f
2748:t
2744:t
2742:(
2740:f
2722:t
2714:t
2712:(
2710:f
2702:f
2673:(
2633:(
2620:V
2616:c
2612:Y
2608:V
2604:X
2600:V
2598:(
2596:f
2592:Y
2588:X
2584:f
2559:(
2539:(
2531:)
2527:(
2510:.
2482:μ
2474:σ
2462:μ
2457:X
2451:μ
2437:μ
2430:Y
2426:X
2417:μ
2407:μ
2402:f
2389:μ
2382:μ
2376:μ
2370:Y
2366:X
2364::
2362:f
2356:Y
2351:μ
2346:X
2336:μ
2318:(
2297:(
2291:B
2277:)
2274:B
2271:(
2266:1
2259:f
2248:a
2234:)
2231:)
2225:,
2222:x
2219:(
2216:(
2211:1
2204:f
2179:(
2154:(
2123:)
2115:(
2110:k
2106:y
2101:k
2097:y
2093:k
2089:k
2086:y
2084:(
2082:f
2062:(
2056:2
2053:y
2051:(
2049:f
2045:1
2042:y
2040:(
2038:f
2034:2
2031:y
2027:1
2024:y
2020:f
2000:)
1992:(
1988:Q
1984:Q
1980:Q
1976:x
1974:(
1972:f
1968:x
1964:f
1960:Q
1947:(
1941:Z
1933:f
1925:Z
1921:f
1917:Σ
1913:Z
1909:T
1905:Y
1901:Z
1893:Y
1889:X
1885:f
1841:)
1836:2
1832:)
1828:g
1822:f
1819:(
1816:+
1811:2
1807:)
1803:g
1800:+
1797:f
1794:(
1791:(
1786:4
1783:1
1778:=
1775:g
1772:f
1750:g
1747:+
1744:f
1722:g
1719:,
1716:f
1692:k
1688:f
1665:f
1629:g
1627:+
1625:f
1598:+
1344:]
1341:E
1338:[
1333:1
1326:g
1322:=
1319:]
1316:]
1313:D
1310:[
1305:1
1298:h
1294:[
1289:1
1282:g
1278:=
1275:]
1272:D
1269:[
1264:1
1257:)
1253:g
1247:h
1244:(
1224:]
1221:D
1218:[
1213:1
1206:h
1199:E
1189:Z
1185:D
1181:Y
1177:σ
1163:Z
1157:Y
1154::
1151:h
1131:Y
1125:X
1122::
1119:g
1086:A
1082:B
1078:f
1074:A
1058:B
1055:=
1052:]
1049:A
1046:[
1041:1
1034:g
1030:=
1027:]
1024:]
1021:)
1015:,
1012:0
1009:(
1006:[
1001:1
994:h
990:[
985:1
978:g
957:)
951:,
948:0
945:(
925:g
919:h
909:A
905:h
898:A
894:C
890:g
886:A
882:F
878:B
871:C
867:f
863:F
844:1
837:f
830:g
820:f
810:C
802:1
799:f
795:x
793:(
791:1
788:f
784:x
780:x
778:(
776:f
772:f
734:f
730:f
726:R
722:R
718:f
658:Y
654:X
601:X
495:C
489:B
464:R
458:B
435:)
430:C
424:B
418:,
413:M
408:(
388:)
383:R
377:B
371:,
366:M
361:(
332:M
320:X
316:f
302:)
297:M
292:,
289:X
286:(
179:)
143:.
42::
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
