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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58 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
33 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
48 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
60 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
63 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
40 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
98 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
31 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
95 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
43 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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36 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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215 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
81 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
94 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 ...
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2answers
149 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
64 views

Finding limit of $p$-norm of succession of functions

Let $\,f,g \in L^{p} ( \mathbb R^{n} ,\mathcal{L}, m)$, $\mathcal{L}$ being the $\sigma$-algebra of Lebesgue-measurable sets and $m$ being the Lebesgue measure on $\mathbb R^{n}$. Now, for $1\le p ...
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1answer
117 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
35 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
60 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
53 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
45 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: ...
5
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0answers
65 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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99 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
61 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
40 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
97 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 ...
3
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1answer
144 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
29 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
53 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
40 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
56 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
51 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
36 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
41 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
37 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 ...
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1answer
41 views

On $C_c^{\infty}$ being dense in $L^p$

We had the theorem about $C_c^{\infty}$ being dense in $L^p$, which, as I understand, means that if we already have an $L^p$ function, there is a $C_c^{\infty}$ function arbitrary close to it with ...
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1answer
64 views

Integration of standard multivariate normal distribution

We should express the integral $I_{n}=\int_{\mathbb{R}^{n}}\exp\left(\frac{-\left\Vert x\right\Vert ^{2}}{2}\right)\mathrm{d}x$ using $I_1$. Where $\left\Vert x\right\Vert =\left(x_{1}^{2}+\cdots ...
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0answers
35 views

lebesgue integral of $f(x^n)$

I know that $f:[0,1]\to \mathbb{R}$ is continuous at $0$, and $f\in L_1([0,1])$. How can one prove that $f(x^n)\in L_1([0,1])$, for any $n\in \mathbb{N}$?
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1answer
71 views

Why is the topology of convergence in measure equivalent to this metric here?

I am currently struggeling with the topic of convergence in measure topologies. Now I read that on the space of measurable function $L^0$ on $[0,1]$ with the Borel sigma algebra and the lebesgue ...
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2answers
57 views

Lebesgue integral of a non-negative function.

I have been looking at Kolmogorov's book "Introductory Real Analysis" and have become stuck at the problem 4a) on page 301. In this problem we are given $f$ a nonnegative and integrable function on ...
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1answer
27 views

For which a and b is $\int_0^{1/2} r^{a+n-1}|\log(r)|^b dr<\infty$?

The problem I am working on asks which real values of a and b make $|x|^a|\log|x||^b$ integrable over $\{x \in \mathbb{R}: |x| < 1/2\}$, but I reinterpret the question to asking which real values ...
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1answer
100 views

f being a Lebesgue integrable function on $(0, a)$ implies that $g(x) = \int_x^a (f(t)/t)dt$ is also integrable.

I need to prove: If f is Lebesgue integrable on $(0, a)$ and $g(x) = \int_x^a (f(t)/t)dt$, then g is integrable on $(0, a)$. I know that since f is integrable on the interval $(0, a)$ I have ...
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2answers
136 views

Biorthogonal functions in $L^p$

I asked one question that is already answered: 1.) I have a question about Lemma 9.5 on page 93/94 reference. It's about the part of the proof where the sequence of $(g_n^*)$ are introduced. I don't ...
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0answers
48 views

What means: is equivalent to?

I found the following theorem: Let $(f_n)$ be a sequence of norm one functions in $L^p, p \in [1, \infty)$. If $\lambda(supp(f_n)) \rightarrow 0$, then some subsequence of $(f_n)$ is equivalent to a ...
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1answer
37 views

Weak absolute continuity of measures

I want to show that if we have a function $f \in L^p$ sucht that $||f||_p =1$. Define a new measure $\mu$ by $$\mu(A):=\int_A |f(x)|^p dm(x).$$ Then $\forall \epsilon > 0 \ \ \exists ...
1
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1answer
37 views

Sobolev inequality in negative index

For $s>n/2$, is it true that $$ \int |fg| dx\leq ||f||_{H^s}||g||_{H^{-s}}?$$ This inequality is used on pg 398 of the Majda Bertozzi book on Vorticity and Incompressible flow but I can't make ...