# Tagged Questions

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### If $\mu(X)<\infty$, $\int |f_n|<C$ for all n, and $f_n\rightarrow f$ a.e.Show that $f_n\rightarrow f$ in $L^1$.

If $\mu(X)<\infty$, $\int |f_n|<C$ for all n, and $f_n\rightarrow f$ a.e.Show that $f_n\rightarrow f$ in $L^1$. I tried to use uniform integrability but I could not figure out completely. ...
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### Properties of Lebesgue Integration

I am completely stuck with the following problem on Lebesgue Integration: Let $f:\mathbb{R}^d \to [0, +\infty]$ be measurable. Show that if $\int_{\mathbb{R}^d} f(x)dx < \infty$, then $f$ is ...
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### Elias Stein : Real Analysis

I cannot understand why this particular line in the text is true: " Moreover, there are $O(k^{d-1})$ cubes in $\cal{Q}\ '$ " For the text see ...
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### Does this proof for the MCT hold for the extended real valued functions.

Here is a proof for the MCT, but it says that it is for the real numbers, not the extended real numbers. If we allow the function f to take the value infinity does the proof still hold? I can not see ...
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### Can simple functions take the value infinity?

I don't think my book is clear about this. It is "a course in real analysis", by weiss. Now I am in the chapter about the general lebesge integral, and we are going to develop the non-negative ...
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### Can this exercise be solved by DCT, I was only able to use MCT.

How would you solve this exercise? You don't need to give me the details, just the general idea. Let f be a Lebesgue integrable function. Show that $\int f(x+a) d\lambda=f(x) d\lambda$ and ...
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### Calculate Radon-Nikodym derivative in a point when it is continuous in that point

I can't solve the following exercise, even if I find it quite intuitive. Let $\nu, \mu$ be Radon measures on a metric space $(X,d)$. Suppose that: 1) $w\in L^1(X,\mu), w\geq 0$ $\mu$ a.e.; 2) $w$ is ...
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### can representatives of an equivalence class of L_1(r^d)be nonmeasurable

Motivation: The composition of 2 Lebesgue measurable functions need not be measurable. This problem can be dealt with in a case by case basis( like with convolutions). Or as Big Rudin does, apply ...
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### Prove something is a signed measure

Given a measure space $(X,\mathcal{M},\mu)$ and a measurable function $f:X\rightarrow \overline{\mathbb{R}}$ such that at least one of $f^+$ or $f^-$ is integrable, show that ...
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### Non Borel Spaces: Gauge Integral

Is there a gauge integral over non Borel spaces? (My interest lies in finite measure spaces.) It seems as the definition of the gauge integral crucially depends on the existence of open sets for a ...
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### Riemann implies Lebesgue integrablility on $\mathbb{R}^n$, prove $f(x)$ continuous at x where $g(x)=G(x)$

Let $f:[a_1,b_1]\times \cdots \times[a_n,b_n] \rightarrow \mathbb{R}$ be Riemann integrable. Prove that is $f$ Lebesgue integrable. Proof: $$Q:= [a_1,b_1]\times \cdots \times [a_n,b_n].$$ For simple ...
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### A tricky integral with vanishing domain

I would love to have the following result, however I got no clue if it is even true! Let $B_n:=\{y:\varepsilon_n<|y|\leq\tilde{\varepsilon}_n\}$ for some sequences ...
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### Why is the value assigned to a gauge integral well defined (unique)?

Why is the value assigned to a gauge integral well defined (unique)? If we would have given a net (so an underlying order that happens to be directed), then the limit would be unique given a ...