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Struggling with a university math problem, I'd be interested if there was actually someone who can solve it.
Let m be a measure on a set X, and let {An}_n
be a sequence of subsets of X satisfying
m(A1)+m(A2)+... finite
Consider the set
E = {x element of X : x belongs to An for infinitely many n}
show that m(E) = 0.

Someone on HLTV I should add

1 reply

yes

What math level is this bro wtf is going on

3 replies

Bachelor Mathematics

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😧
If m(A1)+m(A2)... Is finite doesnt that mean that each segment must = 0? Otherwise the sum would be infinite

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yes

I know the answer

is that sum of the measures equation right? isnt the sum supposed to be zero or something
also if you're being serious u should probably try math stack exchange

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+1 questions almost always get answered there. Idk what this problem is

i know the answer but if i tell you wont learn

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+1

Answer is quite obvious dont you think?

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+1 I don't get his problem

Need help for middle school math lmao

im so glad im done with this witchery

Here's a hint. E is also called the limsup of An. You can show its equal to intersection(n=1:inf) union (k=n:inf) An. If you haven't proven that already, you should be able to show it yourself.
You can use countable subadditivity and that fact that if an infinite series converges, the elements must converge to 0 to complete the proof. I'll leave the details to you, because you won't pass your class otherwise.

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Basic idea: Sum{An}_n is finite means there is some sort of convergence hence there is only finitely many {An} with m{An}>1/N for all N that means x is in "Infinitely" small {An}'s keep this and it follow's m(E) has to be 0

3rd Semester Maß und Integrationsthorie?

2 replies

Yep, ETH Zurich

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