Tagged Questions

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Exercise on Dominated convergence theorem

Consider the sequence $f_n=(-1)^n \frac{x}{\log(1+x)} \chi_{(0,1/n)}(x)$. Is it true that $$\sum_n \int_X f_n d\mu= \int_X \sum_n f_n d\mu$$ with $X=(0,1)$? I was thinking about using the ...
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Do simple functions converge almost everywhere?

Assume there is a sequence of simple functions s.t.: $$\|\int(s_m - s_n)\mathrm{d}\mu\|\to 0$$ Does it follow that there is a subsequence which converges almost everywhere? (Note the order of modulus ...
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Does uniform integrability plus convergence in measure imply convergence in $L^1$?

Does uniform integrability plus convergence in measure imply convergence in $L^1$? I know this holds on a probability space. Does it hold on a general measure space? I have tried googling. It ...
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Example of an integral not converging

Consider a measurable space $(\Omega,\mathcal{A},\mu)$ with $\mu(\Omega)<\infty$. Let $f_1,f_2,\ldots$ be bounded measurable functions so, that $f_n\to f$ uniformly. Then $f$ is measurable ...
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Lebesgue's Dominated Convergence Theorem problem

I am having trouble using DCT for the following Prove $$\lim_{n\to\infty}\int_0^\infty \frac{n}{(1+y)^n(ny)^\frac{1}{n}}dy = 1$$ I think most of the mass of the integral lies beneath $(e-1)/n$ but ...
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Parameter integral and continuity (Theorem of Lebesgue)

I already kept myself busy with a proof concerning the Theorem of Lebesgue and differentiation of a parameter integral. Unfortunately I did not get an answer there yet. Now my task is nearly the ...
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Uniform convergence and convergence of integrals

Question: $(X,\mathcal{M},\mu)$ measure space. Suppose that $\{f_n\}\subset L^1$ and $f_n\rightarrow f$ uniformly. Show that if $\mu(X) <\infty$, then $\int f_n\rightarrow \int f$. Proof: I ...
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Theorem of Lebesgue and differentiation of a parameter integral

Let $(a,b)\subset\mathbb{R}$ be an interval and let $\left\{f_t\colon\Omega\to\mathbb{R}\right\}_{t\in (a,b)}$ be a family of measurable functions on the measurable space ...
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Integration by part for Lebesgue integral

I tried to prove one theorem in convergence of random variables and found myself in a little bit of trouble when doing integration by part. The reason being it involves a Lebesgue integral which I am ...
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Prove convergence of improper integral using change of variable.

This may be trivial, but I could use some help... Consider a real function $f: (0,1) \rightarrow \mathbb{R}$, continuous, positive, but not necessarily bounded. Let $g: [0,1] \rightarrow [0,1]$ be a ...
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Convergence and Lebesgue Integration

I came across this question in a textbook on introductory Lebesgue Integration. I have been teaching myself this material but was unsure of how to do the following question: Let $(g_n)$ be a sequence ...
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The General Lebesgue Integral

For a measurable function, $f$, on $[1, \infty)$ which is bounded on bounded sets, define $a_n = \int_n^{n+1} f$ for each natural number $n$. Is it true that $f$ is integrable over $[1, \infty)$ if ...
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{$\int_{[1/n,1]}f$} to converge and yet $f$ is not $L$-integrable over $[0,1]$

Let $f$ be a function on $[0,1]$ and continuous on $(0,1]$. I want to find a function $f$ s.t. {$\int_{[1/n,1]}f$} converges and yet $f$ is not $L$-integrable over $[0,1]$. My attempts: I've found ...
I have problem with the exercise that follows. Let $(z_m)_m \in R^n$ so that $\Vert z_m \Vert \rightarrow \infty$ when $m\to \infty$. Let $f:R^n \rightarrow [-\infty;+\infty]$ integrable. Show ...
Is what I'm doing valid if we don't have any information on boundedness of $f$ or $f_n$? let $X$ be a finite measure space and $\{f_n\}$ be a sequence of nonnegative integrable functions, \$f_n ...