Graduate Real Analysis..panic

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saus
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Graduate Real Analysis..panic

Postby saus » Mon Oct 10, 2011 6:21 am UTC

I have a test coming up on sigma-algebras, measures, caratheodory's theorem, Borel/Lebesgue measure, integration, monotone convergence/fatou/dominated convergence, product measures/fubini-tonelli, integration in polar coordinates (first 2 chapters in Folland's book). I understand everything fine and I can follow and maybe even reproduce the proofs given in the book, but I generally have no idea what to do on the homework (exercises in the book). I went to my professor and he said I should play around more. This resulted in me not turning in my homework more often than not. And I certainly don't want the test to end up like my homework. I guess finding a solution doesnt feel at all intuitive most of the time. I just don't know how to acquire this skill. I keep telling myself I'll go to the prof for help, but just the way he said I should play around more it seems like he doesn't really want me to come. I mean it's true you never really learn math until you work out a problem yourself but that just isn't happening. I want to do practice problems to prepare for the test, but I can't get anywhere most of the time, unless I spend hours thinking about a single problem, which won't work for a 75 minute test. Maybe the test problems will be easier, but I still am left with the problem of not being able to do my homework. </rant>

So where do I go from here?

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Re: Graduate Real Analysis..panic

Postby jestingrabbit » Mon Oct 10, 2011 6:49 am UTC

If you write out some of the problems that you're having problems with it might be possible to talk through solutions, and maybe from there you might wrap your head around a process to pursue.
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Re: Graduate Real Analysis..panic

Postby skeptical scientist » Mon Oct 10, 2011 8:30 am UTC

saus wrote:I want to do practice problems to prepare for the test, but I can't get anywhere most of the time, unless I spend hours thinking about a single problem, which won't work for a 75 minute test.?

The best suggestion I have is just do the practice problems, even if they take an hour or two each at first. After you do several, they should be easier and faster and thus reasonable for a timed exam.
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Re: Graduate Real Analysis..panic

Postby Yakk » Mon Oct 10, 2011 7:45 pm UTC

I've easily spent hours on a single question in a proof-based course.

Generally in coursework, your proofs are going to have patterns. You'll have been expected to know patterns from prior courses, and new kinds of proofs will show up in this new material.

How heavy is your prior proof course work experience? What kinds of thought patterns do you follow when trying to prove something?
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saus
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Re: Graduate Real Analysis..panic

Postby saus » Mon Oct 10, 2011 8:54 pm UTC

Proofs I had to create for my undergrad courses were all very obvious as long as you knew the definitions.

This time around my assault on some problems is going well. I think I just gleaned general techniques from the proofs given in the book that maybe I didn't internalize before.

A problem I'm stuck on:
-(X,M) is a measurable space. If [imath]\{f_n\}[/imath] is a sequence of (real valued) measurable functions on X, then [imath]E=\{x:\lim f_n (x) exists\}[/imath] is a measurable set.

So my strategy is something like this: Let [imath]f(x)=\lim f_n (x)[/imath]. The union of intervals of length [imath]\epsilon[/imath] around f(x) for all x in E is an open set, so it's a borel set.

For each x in E, there is an N where if n>N then fn(x) is close to f(x). The problem is there's possibly uncountably many of these N's so I can't take the max of them...

What I would like to do is basically show the preimage of the union of all these epsilon wide intervals is in M because the fn's are measurable, so a countable intersection that is just the set E is in M. I'm not sure if I could somehow take a finite number of these N's and do a limiting process to get what I want..

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Re: Graduate Real Analysis..panic

Postby imatrendytotebag » Mon Oct 10, 2011 10:00 pm UTC

Since there are only so many test questions that can be asked about this material, the more problems you do the more likely it is that the test problems are very similar to problems you've already done.

For the problem you gave about the sequence of functions, I have two ideas. The first is, remember that the countable union of measurable sets is measurable: The set you want to prove is measurable might be better viewed as a countable union of simpler sets which can more easily be seen to be measurable. I don't know exactly how it would work in your case, but remember that you can take epsilon in the definition of limit to be rational without any consequences.

The second hint (which points toward a different solution method than what you were attempting) is to, when given a series of functions, remember other measurable functions you can build out of them: inf, sup, lim inf, lim sup, etc. What is a criteria for the sequence fn(x) having a limit (that does not depend on the value of the limit)?
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Re: Graduate Real Analysis..panic

Postby saus » Mon Oct 10, 2011 11:16 pm UTC

So lim sup fn and lim inf fn are measurable, then g(x) = limsup fn - liminf fn is measurable, so with {0} a borel set, g^-1({0}) is a measurable set, which is the set of x where limsup fn=liminf fn, that is, the set where lim fn exists.

This is what is frustrating about doing these problems for the first time. I can brute force myself way through an ugly proof and possibly get nowhere, without even considering a much simpler solution (assuming there arent any mistakes in the above).

Next up before I move on to integrals + convergence thms: [imath]\mu[/imath] is complete iff ((fn measurable and fn->f almost everywhere) implies f is measurable).
fn:(X,M)->(Y,N)
(=>) lim inf fn = f except on some set A of measure zero. For E in N, then f^-1(E) = (lim inf fn)^-1(E) U B, where B is contained in A (proof by a picture I drew ..?). The second half is the union of 2 elements of M, f is measurable. Something like that.

(<=) I feel like I need to construct some measurable fn whose limit disagrees on the union of all null sets with some f. Then yknow.. subtracting the fn sets from the f sets to get arbitrary subsets of null sets to be measurable.
Last edited by saus on Tue Oct 11, 2011 12:08 am UTC, edited 1 time in total.

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Re: Graduate Real Analysis..panic

Postby theodds » Tue Oct 11, 2011 12:00 am UTC

You need to make sure that the arithmetic in [imath]g(x) = \limsup f_n(x) - \liminf f_n(x)[/imath] is well defined, i.e. you don't want [imath]\infty - \infty[/imath] but it's easy to patch this up.

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Re: Graduate Real Analysis..panic

Postby saus » Tue Oct 11, 2011 12:11 am UTC

So how do I do that without messing up the measurability of g. The set that this occurs on is measurable..because it's lim sup fn^-1({infinity}) intersect lim inf fn^-1({infinity})..
So if I set the value of g(x)=1 for x in this set, it should still be measurable?
Last edited by saus on Tue Oct 11, 2011 12:15 am UTC, edited 1 time in total.

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Re: Graduate Real Analysis..panic

Postby jestingrabbit » Tue Oct 11, 2011 12:25 am UTC

saus wrote:So how do I do that without messing up the measurability of g. The set that this occurs on is measurable..because it's lim sup fn^-1({infinity}) intersect lim inf fn^-1({infinity})..
So if I set the value of g(x)=1 for x in this set, it should still be measurable?


Yes.
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Re: Graduate Real Analysis..panic

Postby Yakk » Tue Oct 11, 2011 12:26 am UTC

Or, if f_sup:R->RU{inf}, and f_inf:R->RU{-inf}, define A = f_sup^-1(R) intersect f_inf^-1(R), and reason with the domain of A instead of R?
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Re: Graduate Real Analysis..panic

Postby Yakk » Tue Oct 11, 2011 12:32 am UTC

For <= I'd be tempted to make mu incomplete (ie, there is a subset of a null set that is immeasurable), take a sequence f_n that converges a.e. to some function f, and arrange it so that f isn't measurable.

Why? Because an "almost complete" mu should be sufficient to show this property. So if I make a mu that is "artificially incomplete", it should be enough to cause a problem.

So my first attempt at grasping what goes wrong is to start with a PARTICULAR measure mu that is complete, make it incomplete "just barely", then build a f_n that converges to f almost everywhere but f isn't measurable. This doesn't prove the contrapositive -- but it might provide a route to make a more abstract proof to do so.

How do I make it "just barely" incomplete? I'd play around.
One of the painful things about our time is that those who feel certainty are stupid, and those with any imagination and understanding are filled with doubt and indecision - BR

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saus
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Re: Graduate Real Analysis..panic

Postby saus » Wed Oct 12, 2011 3:01 am UTC

For the previous problem about complete measures.. I kinda get it but I feel like I need to move on, I'll ask a friend about it maybe.

-If f is in L+ and the integral of f is finite, then E={x:f(x)=infinity} is a null set and F={x: f(x)>0} is sigma-finite.

For the null set one: Assume the set had nonzero measure. There exists a sequence of simple functions converging pointwise to f, apply monotone convergence theorem to get the integral of f is the limit of the integrals of the simple functions, but the limit of the integral of the simple functions is unbounded on E. Then the integral of f would be infinite.

For the next part I'm going to try something similar to the proof in the book of another theorem. Consider Fk={x: f(x)>1/k}. These sets must have finite measure, or else f would have an infinite integral similarly to the above. And the (countable) union of all Fk is F. f is measurable so the preimage of (1/k, infinity] is a measurable set, which are the Fk. Is that it?

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Re: Graduate Real Analysis..panic

Postby Yakk » Wed Oct 12, 2011 3:16 am UTC

Yes. Well, assuming L+ is what I gather it to be given how you used it. :)

If I was feeling paranoid, I'd show that the countable union of F_k is F, but that would be paranoia speaking.
One of the painful things about our time is that those who feel certainty are stupid, and those with any imagination and understanding are filled with doubt and indecision - BR

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saus
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Re: Graduate Real Analysis..panic

Postby saus » Wed Oct 12, 2011 3:34 am UTC

L+ is folland's notation for measurable functions into [0,infinity].
x is in the union of Fk iff f(x) > 1/k for all natural #s k iff f(x)>0 (contrapositives are easy to check for final iff).

Next: Assume Fatou's lemma and deduce the monotone convergence theorem from it.

So we have an increasing sequence of fn's whose limit is f. We assume f=lim fn, so lim inf of fn is f, and applying fatou's lemma id get [imath]\int f \leq \liminf \int f_n[/imath]. I want to get to [imath]\int f = \lim \int f_n[/imath]. Perhaps I can show the reverse inequality and the sup part trivally. Will think about it later.

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Re: Graduate Real Analysis..panic

Postby Yakk » Wed Oct 12, 2011 4:08 am UTC

Try a concrete, simple example. What happens when you assume f isn't equal to that limit?

Can you generalize?
One of the painful things about our time is that those who feel certainty are stupid, and those with any imagination and understanding are filled with doubt and indecision - BR

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Re: Graduate Real Analysis..panic

Postby saus » Thu Oct 13, 2011 7:45 pm UTC

I figured out MCT=>Fatou last night. Let's see, integral of fn is <= integral of f for all n, so sup int fn <= int f, and limsup int fn is the same as the sup because the fn are increasing. So int f is between a lim sup and a lim inf, everything ends up being equal.

Even better, the test was today and I answered everything I needed to, to my knowledge completely and correctly. Thanks sooo much everyone who helped me, I no longer have to feel like it's only a matter of time before I drop out (for now).

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Re: Graduate Real Analysis..panic

Postby Yakk » Thu Oct 13, 2011 9:27 pm UTC

A more concrete method is taking a look at the definition of the integral. Suppose int f is less than int f_n for some n by at least epsilon > 0. Then we can find a sum of simple functions whose int is within epsilon/2 of f_n by the definition of what integral of f_n means. As f_n is also strictly less than f, these functions are also under f. Which means that epsilon/2 is less than, or equal to, zero.
One of the painful things about our time is that those who feel certainty are stupid, and those with any imagination and understanding are filled with doubt and indecision - BR

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