For question about integration, where the theory is based on measures. So it's almost always used together with the tag [measure-theory], and its aim is to specify questions about integral, not only properties of the measure.

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33 views

Monotone convergence, measure-theory, is this excercise correct?

Here is the exercise: I have some questions: Is this correct when k starts with 1?, the Taylor series with e starts with 0? But does the zero disappear in some way?, I can not see how. I know that ...
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1answer
86 views

Riemann-Lebesgue lemma

How can I prove the following result? Let $([-1,1],M,m)$ a measure space, where $m$ is the Lebesgue measure in $[-1,1]$. If $f$ is Lebesgue integrable, then ...
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1answer
32 views

Functions in $L^p$ and $L^q$ spaces

For any two different numbers $p,q\in[1,\infty)$ find functions $f\in L^p \setminus L^q$ and $g\in L^q \setminus L^p$. Solution: let $$f(x)=x^{-1/p}(1+|\log x|)^{-2/p}$$ Then $$\int|f|^p = ...
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1answer
69 views

Showing $\frac{\log(x)}{1+x^2}$ is Lebesgue integrable

I have calculated the integral: $\int_0^\infty \frac{\log(x)}{1+x^2} dx$ using a contour integral. However I was wondering how we would show that this is Lebesgue integrable. I have thought about ...
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1answer
47 views

Integrate over different measures

In Probability theory the expected value of a random variables $X : \Omega \rightarrow \mathbb{R}$ is defined as $E(X) = \int_\Omega X dP$ Now, if $\Omega \subset \mathbb{R}$ and has a density ...
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2answers
61 views

Why is $L^p$ isomorphic to $(L^p)^2$

Is it possible to say why the spaces in the title are isomorphic as Banach spaces? Is their a Theorem that says this or is it even possible to find an explicit representation of this isomorphism?
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37 views

Not measurable function whose module is measurable

I read through my notes that is trivial to find a not measurable function $f$ whose module $|f|$ is measurable. However I don't know how to provide such an example.
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1answer
90 views

Exercise on measure theory

Let $X\neq \emptyset$ and $f:X \rightarrow [0, \infty]$ not identical infinity. Set $$ \sum_{x \in X} f(x)= \sup \left\{ \sum_{x \in F}f(x), F \subseteq X, F \mbox{ finite} \right\}.$$ $(i)$ Show ...
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1answer
49 views

Equivalent ideas of absolute continuity of measures

Wikipedia says that $\mu$ is absolutely continuous with respect to $\nu$, if $\nu(A)=0 \Rightarrow \mu(A)=0$. Okay, then I found another notion of absolute continuous measures: Let $||f||_1=1$ and ...
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1answer
37 views

Convex , then also Measurable

I was reading about Jensen's inequality and noticed that don't require $\phi$ to be measurable here: Wikipedia link. Therefore, I guess that being convex implies being measurable somehow, but I have ...
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2answers
37 views

Measurability of f(x) to g(x)

Let $(\Omega, S)$ be a measurable space. If $f:\Omega \rightarrow \mathbb{R}$ is a strictly positive measurable function and $g:\Omega \rightarrow \mathbb{R}$ is measurable show that $f^g$ is ...
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1answer
52 views

Computing limit with Dominated Convergence Theorem

I am trying to compute the following limit: $$ \lim_{n\to \infty} \int_0^\infty \frac{x^{n-2}}{1+x^n} \cos(\pi n x) \,dx . $$ This is a problem in an old analysis qualifying exam. Let $f_n(x) = ...
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1answer
52 views

Continuity of a function defined by means of the Lebesgue measure

Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a Lebesgue measurable function and $\phi(x)=\lambda ( \lbrace{ t: f(t) >x \rbrace} )$. Prove that $\phi$ is right-continuous but not necessarily ...
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1answer
29 views

Reason for product sigma algebra notation

I was wondering why it is so common to denote the product sigma algebra with the same symbol that is used for tensor products. Is there a specific reason that this product symbol is used or was is ...
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0answers
39 views

Right-continuity of functions associated to measures

I would like to show that it's possible to associate to a measure a monotone increasing right-continuous function s.t.: $\mu(\left(a,b\right])=F(b)-F(a)$. How can I prove that a function like ...
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3answers
55 views

property of Lebesgue integral

If $f$ and $g$ are nonnegative Lebesgue measurable functions, then we know that $\int (f+g) d\lambda = \int f d \lambda + \int g d \lambda $. Given the difinition of integral of an arbitrary Lebesgue ...
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1answer
43 views

equality of lim sup and lim

If we know $f_n \rightarrow f$ almost everywhere and if we have :- $$\limsup_{n \to \infty} \int_X |f_n - f|^p \, d\mu \le 0 $$ How can we get $$\lim_{n \to \infty} \int_X |f_n - f|^p \, d\mu = 0$$ ...
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1answer
37 views

What is the largest class of measurable functions $f$ s.t. $f'$ a.e.?

We know by Lebesgue Theorem that monotone functions on interval [a,b] has finite derivate almost everywhere and different of two monotone functions have finite derivative a.e. $\textbf{My Question}$ ...
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1answer
55 views

$L^p$-space inclusions

Let $1\leq p<q<\infty$. Which of the following inclusions are true? $L^p(0,1)\subset L^q(0,1)$ $L^q(0,1)\subset L^p(0,1)$ $L^p(0,\infty)\subset L^q(0,\infty)$ $L^q(0,\infty)\subset ...
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1answer
30 views

Stair-case functions and upper-limit functions

I am trying to come up with a proof of the following proposition: Let $I\subset\mathbb{R}^n$ be an interval and let $(s_k)$ be a succession of staircase functions defined in $I$, increasing almost ...
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2answers
87 views

How does Wikipedia's definition of the Lebesgue integral relate to more common definitions?

Wikipedia presents a definition of the Lebesgue integral (of a nonnegative function $f$) that I hadn't encountered before: Let $f^*(t)=\mu \left (\{x\mid f(x)>t\} \right )$. The Lebesgue ...
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1answer
32 views

Is the function $f(x,y) = \frac{x-y}{(x + y)^3}$ lebesgue integrable.

I'm trying to show whether the function $f(x,y) = \frac{x-y}{(x + y)^3}$ is Lebesgue integrable on $[0,1]\times[0,1]$. I've split the region into two parts $x>y$ and $x<y$ (by the symmetry of ...
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1answer
30 views

Lebesgue measure unique on semiring?

in our lecture it was stated that the Lebesgue measure can be uniquely extended from a semiring to a sigma algebra by Caratheodory's theorem. Unfortunately, we did not show that it is unique on the ...
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0answers
90 views

Generate Borel Sigma Algebra

I want to show that the Borel Sigma-Algebra on $\mathbb{R}^n$ is generated by $ A:= \{(a_1,b_1] \times \cdots\times (a_n,b_n]; a_i,b_i \in \mathbb{R} \}$ as well as $ B:= \{(-\infty,c_1] ...
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2answers
79 views

Show the function is integrable and find the integral - somewhat complex question

We are given $Q = [0,1]$x$[0,1]$ We are also given the function $f(x,y) = (\frac{1}{10})^n$ where $\frac{1}{2^{n+1}} < \max(x,y) \leq \frac{1}{2^n}, (n=0,1,2,...)$ and $f(0,0)=0$. Show that $f$ ...
4
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1answer
67 views

For what values of $\alpha,\beta$ is $x^{\alpha}\sin{x^\beta}\in L^1((0,1])$?

Let $E=(0,1]$. For every $\alpha,\beta\in\mathbb{R}$, let $f(x)=x^{\alpha}\sin{x^\beta}$. For what values of $\alpha,\beta$ is $f\in L^1(E)$? I think I know the answer: when $\alpha>-1$ or ...
3
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2answers
109 views

What is so special about the Lebesgue-Stieltjes measure

A measure $\lambda: B(\mathbb{R}^n) \rightarrow \overline{{\mathbb{R_{\ge 0}}}}$ that is associated with a monotone increasing and right-side continuous function $F$ is called a Lebesgue-Stieltjes ...
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1answer
113 views

Show $\lim\limits_{a \rightarrow + \infty} \int_0^{\infty} \frac{1}{1+y^2}e^{-ay} dy =0 $

Need to prove $\lim\limits_{a \rightarrow + \infty} \int_0^{\infty} \frac{1}{1+y^2}e^{-ay} dy =0 $ and $\lim\limits_{a \rightarrow + \infty} \int_0^{\infty} \frac{y}{1+y^2}e^{-ay} dy =0 $ Can ...
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1answer
28 views

norm on a quotient-space

Let $M:[0,\infty)\to[0,\infty)$ be continuous and convex. Further $M$ satisfies $M(t)=0\Leftrightarrow t=0$. Let $$\mathcal L_M(\mathbb R):=\left\{f:\mathbb R\to\mathbb R \mathrm{\ measurable\ ...
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1answer
30 views

What is the name of this measure property?

if we have a function $f \in L^p$ sucht that $||f||_p =1$ and $m$ being a finite measure. Define a new measure $\mu$ by $$\mu(A):=\int_A |f(x)|^p dm(x).$$ Then $\forall \epsilon > 0 \ \ ...
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1answer
30 views

An integral of a sequence of functions

$\Omega\subset\mathbb{R}^n$ is a bounded domain with smooth boundary $\partial\Omega$. Does $$ \liminf_{k\rightarrow\infty} \int_{\Omega} \rho(u_k)\,dx \geq \int_{\Omega} \liminf_{k\rightarrow\infty} ...
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3answers
59 views

If $f\in L^1$ has a compact support and $0 \leq p \leq1$ then $|f|^p\in L^1$

My text proved that If $f\in L^1$ is bounded and $p \geq1$ then $|f|^p\in L^1$ I wanted to prove the seemingly very similar statement: If $f\in L^1$ has a compact support and $0 \leq p ...
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1answer
47 views

Is this space complete?

Let $X$ be the space of measurable functions $f:[0,1] \rightarrow \mathbb{R}$. I want to find out whether this space is complete under the metric $d(f,g):= \int_{[0,1]} \frac{|f-g|}{1 + |f-g|}$. Does ...
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0answers
28 views

Under which assumptions we have $f\in L^p$ for all $p\in\mathbb N$

So here is my question, I wanted to generalize, under what assumptions for some $f$ we have $f\in L^p(\mathbb R)\;\forall p\in\mathbb N.$ And I found the following, Let $f\in L^p(\mathbb R)$ for ...
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0answers
41 views

Higher-dimension integrability (over rectangles) well-defined

Here is the problem and my work toward a proof: Question: Prove that in the following definition, the value of $\int_E f dx$ is independent of the choice of rectangle $J$: Definition: ...
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52 views

Looking for a a measure-theoretic treatment of “differential entropy”

If $X$ is a discrete random variable, its entropy $H(X)$ is usually defined as something along the lines of $-\sum \def\P{\mathbb{P}}\P(x) \log_2( \P(x))$, where the sum ranges over all the possible ...
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0answers
75 views

$\lim\limits_{n\to\infty}\displaystyle\int_X n\log((1+(f/n)^{\alpha})d\mu$

suppose $\mu$ is a positive measure on $X$ and $f:X\to[0,\infty]$ is measurable with $\int_Xfd\mu=c$, where $0<c<\infty$ and let $\alpha$ be a constant, prove that; ...
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1answer
57 views

Show that $\frac{1}{x^4 \sin^2 (x) +1} \in L^1([0, \infty))$

This is question 10.20c from Apostol's Mathematical Analysis. Basically, I am trying to show that $$ f(x)=\frac{1}{x^4 \sin^2(x)+1} \in L^1([0,\infty)) $$ I know that for some value $k$, I can ...
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1answer
39 views

Integral convergence

Please how can I show that the $\lim_{n \rightarrow \infty}\int_{\Re^+}f_n d\mu$ converges and determine its limit in the following cases of $f_n: \Re^+ \rightarrow \Re$ (a)$f_n(x) = sin(nx) ...
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2answers
78 views

What does it mean to be an L^1 function?

I am struggling to understand what the space L^1 is, and what it means for a function to be L^1. A friend told me that a function f is $L^1$ if $\int_\mathbb{R} |f|$ is finite. It is $L^2$ if ...
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1answer
134 views

Function coincides with a function of bounded variation almost everywhere

Problem Suppose $f\in L^1(\mathbb R)$ satisfies that there exists $A\ge0$ such that $$\int_{\mathbb R}\lvert f(x+h)-f(x)\rvert dx\le A\lvert h\rvert$$ for all $h\in\mathbb R$. We need to show that ...
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1answer
24 views

Show the $\min(f_n, f)$ has lebesgue integral converge to $\int_{A}f dm$

let $A$ be a subset of the reals, $f_n,f$ are positive lebesgue measurable functions that $f_n$ converge to f pointwise and $\int_{A}f_n dm$ converge to $\int_{A}f dm$ with $\int_{A}f dm<\infty$. ...
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1answer
49 views

Lebesgue Integrabilityon relative measure space

This shouldn't be too difficult but I just can't get started. There are quite a few similar problems on my problem set so I am hoping to get some idea here on this one, and try solving the others ...
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1answer
38 views

Measure Theory - Convergence of functions with bounded integrals

A question I came across. Let $(X,\mathcal{F},\mu)$ be a $\sigma$ -finite measure space. Let $f_1,f_2,\dotsc:X\to\mathbb R$ be measurable functions such that $n^2\cdot\lVert ...
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0answers
34 views

Measure Theory - Lebesgue Integral over non- $\sigma$-finite spaces

In most courses on Measure Theory the Lebesgue Integral is introduced initially for simple functions on finite spaces, then for general functions on finite spaces and finally for general functions on ...
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0answers
49 views

Measure Theory - An identity for the Lebesgue Intgral

I'm trying to solve the following exercise in Measure Theory: Let $(X,\mathcal{F},\mu)$ be a $\sigma$ -finite measure space. Prove that for every $0\leq f\in L^{1}(\mu)$ it holds that: ...
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1answer
33 views

Extend projection on $L^2$ to one on $L^p$

if we have a closed subspace of $L^p$ called $X \cong l^2$ where the topologies of $L^p$ and $L^2$ coincide (we assume $p>2$). Then we can regard $X$ as a subspace of $L^2$, which means that he is ...
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1answer
30 views

prove that the lebesgue integral of 2 functions is finite

For the first function we have: if $\mu(X)\lt\infty$ and $f\in L^+$ then $\int fd\mu\lt\infty \iff \sum_{n=0}^\infty 2^n\mu(\{x\in X:f(x)\ge2^n\})\lt\infty$ For the second one: let $f\in L^+$ ...
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1answer
35 views

Is $l^p$ closed in $L^p$?

Let's assume we have a subspace $X$ of $L^p$ and we know that $X \cong l^p$(this should just mean isomorph no isometry is assumed here). Can we infer from this that $X$ is closed? I have just read a ...