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Brainlet
Jordan content and measure 2017-02-13 11:51:54 Post No. 8672778
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Jordan content and measure 2017-02-13 11:51:54 Post No. 8672778
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Can any math fag help me out with this? I don't actually want an answer for this problem but merely want to know how long your proof for it is, or if the solution is obvious to you / elegant. I believe I might have found a way to solve this but it basically takes a page or more and I have to restrict myself, in parts, to disjoint open intervals and generalize afterward, which I'm not even sure works since my strategy also implicitly relies on the idea that all [math]\mathcal{I_j}[\math] are subsets of [math][a,b][\math], which does not appear to correct.
The fact that I'm not relying more on (a) an (c) also seems to me to be highly suspicious.
So, are you guys able to prove this in a few lines or in an elegant manner or is your proof also tedious as hell?
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>>8672778
Incidentally, this is not homework but recreational.
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Self indulgent bump.
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Last bump.
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>>8672790
The basic problem is showing that that Jordan measure preserves orderings like we expect(and honestly that's a big thing with lebesgue measure which is kind of ignored).
By this I mean if a set contains another set then its measure is greater than or equal to.
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>>8673296
I don't think showing [math]J(A) \leq J(B)[/math] whenever [math] A \subseteq B [/math] would do anything for (e).
Right now, I'm proving (e) by showing something about the Jordan content of unions of close and open disjoint intervals, which then allows me to state something about their inverse, which in turn allows me to state something about the Jordan content (and the inner Jordan content) of [math]S'[/math], but it's extremely tedious an relies on a proposition such as
>for every family [math]\{I_j\}[/math] of open interval in [math][a,b][/math], there exist a family [math]\{J_i\}[/math] of disjoint open intervals of [math][a,b][/math] such that [math]\bigcup I_j = \bigcup J_i[/math]
I'm just not sure this is the smart way to prove this. It's tedious. I just want to know if someone sees a non tedious way of doing this.
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>>8672778
What book is that from?
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>>8674117
(Pic related)
Which I find excellent in that it really starts from the very beginning. It's an introduction to analysis that starts with taking you by the end and then ends up manifolds and so on.
Still, I gather that nobody has any idea how to solve question (e) elegantly?
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>>8674162
Thanks.
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What are some rigorous books about measure theory?
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