Comments on: 245A, Notes 1: Lebesgue measure
https://terrytao.wordpress.com/2010/09/09/245a-notes-1-lebesgue-measure/
Updates on my research and expository papers, discussion of open problems, and other maths-related topics. By Terence TaoSun, 22 Oct 2017 16:52:06 +0000hourly1http://wordpress.com/By: Terence Tao
https://terrytao.wordpress.com/2010/09/09/245a-notes-1-lebesgue-measure/#comment-485606
Tue, 29 Aug 2017 15:26:39 +0000http://terrytao.wordpress.com/?p=4062#comment-485606Try it! The proof of the monotone class lemma is not exceedingly difficult, and does not require the construction of Lebesgue measure.
]]>By: Terence Tao
https://terrytao.wordpress.com/2010/09/09/245a-notes-1-lebesgue-measure/#comment-485605
Tue, 29 Aug 2017 15:25:56 +0000http://terrytao.wordpress.com/?p=4062#comment-485605Hint: try to think of a set which is “large” but contains no open sets (i.e. has empty interior). Or, conversely, think of a set which is “small” but has huge closure.
]]>By: Anonymous
https://terrytao.wordpress.com/2010/09/09/245a-notes-1-lebesgue-measure/#comment-485557
Tue, 29 Aug 2017 01:01:54 +0000http://terrytao.wordpress.com/?p=4062#comment-485557Exercise 20 is done in Stein-Shakarchi by approximation of open sets. Is it possible to use the monotone class theorem, which is in later notes? Would it be a circular argument?
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https://terrytao.wordpress.com/2010/09/09/245a-notes-1-lebesgue-measure/#comment-485555
Tue, 29 Aug 2017 00:43:22 +0000http://terrytao.wordpress.com/?p=4062#comment-485555I have a dumb question that I don’t know how to answer it myself. The inner regularity and out regularity imply that given any measurable set and , there exists an open set and a closed set such that

and

Is it still true if we switch “open” and “closed” in the statement above? (Do we have approximation by open sets from inside and closed sets from outside?)

]]>By: Terence Tao
https://terrytao.wordpress.com/2010/09/09/245a-notes-1-lebesgue-measure/#comment-485541
Mon, 28 Aug 2017 21:53:26 +0000http://terrytao.wordpress.com/?p=4062#comment-485541If is non-compact then it becomes possible for to be infinite, and so one cannot easily cancel it at the end of the proof of Lemma 10(ii).
]]>By: Anonymous
https://terrytao.wordpress.com/2010/09/09/245a-notes-1-lebesgue-measure/#comment-485449
Sun, 27 Aug 2017 02:38:24 +0000http://terrytao.wordpress.com/?p=4062#comment-485449In the proof of Lemma 10(ii), does one have to reduce to the case when is compact? In order to apply Exercise 5, one only needs one of the sets being compact. On the other hand the finite union of the closed cubes is compact.
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https://terrytao.wordpress.com/2010/09/09/245a-notes-1-lebesgue-measure/#comment-483078
Fri, 23 Jun 2017 21:10:54 +0000http://terrytao.wordpress.com/?p=4062#comment-483078Regarding Exercise 8, is there a reason why we use open sets for the outer approximation and closed sets for the inner approximation but not the other way around (closed sets for the outer and open sets for the inner)? In the Definition 1 of Lebesgue measures, can we use closed containing instead?
]]>By: Anonymous
https://terrytao.wordpress.com/2010/09/09/245a-notes-1-lebesgue-measure/#comment-483062
Fri, 23 Jun 2017 02:14:26 +0000http://terrytao.wordpress.com/?p=4062#comment-483062What would be the trade-off for Definition 1 of Lebesgue measure? As you said under definition 1, Littlewood’s first principle is almost trivial. What’s the cost? What would be the advantage of the one using Carathéodory criterion? How about the one using Riesz representation theorem?
]]>By: Anonymous
https://terrytao.wordpress.com/2010/09/09/245a-notes-1-lebesgue-measure/#comment-478978
Thu, 09 Mar 2017 02:06:23 +0000http://terrytao.wordpress.com/?p=4062#comment-478978I’m trying to understand Axiom 3 and Corollary 4. Without AC, is ZF enough to give the” axiom of finite choice”? For instance suppose is onto, where is finite. Can we *prove* without AC that there is a function such that for each .

[Finite choice can be proven without AC by induction on the cardinality of . -T.]

]]>By: Anonymous
https://terrytao.wordpress.com/2010/09/09/245a-notes-1-lebesgue-measure/#comment-467329
Mon, 28 Mar 2016 19:22:35 +0000http://terrytao.wordpress.com/?p=4062#comment-467329A bounded function on a compact interval is Riemann integrable if and only if it is continuous almost everywhere. Is this true for in general?
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