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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5
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2answers
88 views

Why are some convergent Lebesgue integrals 'undefined'? [duplicate]

I sometimes read statements such as The integral $$\int_0^{\infty} dx \, \frac{\sin x}{x} $$ does not exist as a Lebesgue integral, because it is not absolutely convergent. But according to my ...
2
votes
1answer
75 views

Approximate Identity: Proof?

Problem Given a positive Lebesgue integrable function $j\in\mathcal{L}:j\geq0$. Consider Lebesgue integrable functions $f:\mathbb{R}\to\mathbb{R}$. Then it acts as an approximate identity: ...
1
vote
1answer
29 views

Is the supremum of two-variable measurable function always measurable

Problem : [Let $(X, \mathcal{A})$ and $(Y, \mathcal{B})$ be two measurable spaces and let $f\geq 0$ be measurable with respect to $\mathcal{A} \times \mathcal{B}$. Let $g(x)=\sup_{y\in Y} f(x, y)$ and ...
1
vote
0answers
14 views

$f(x,y)={1 \over x^2} \sum_{n=1}^{\infty}{\int_x^y{\sqrt{t} \over {1+ ({t \over x} -n)^2}}} dt$ is differentiable?

Let $ D=\{(x,y) \in \mathbb{R}^2 : x>0, y>0\}$. Show that the function $$f(x,y)={1 \over x^2} \sum_{n=1}^{\infty}{\int_x^y{\sqrt{t} \over {1+ ({t \over x} -n)^2}}} dt$$ is well defined on $D$. ...
0
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0answers
18 views

What are the intervals $I$ for which the series $\sum_{n=1}^\infty f_n$ converges in norm $L^1 (I)$

For n=1,2,... let $f_n: \mathbb{R} \to \mathbb{R}$ defined by $$f_n(x)= {\sqrt[3]{n-x} \over {e}^{(x-n)^2}}$$ Determine to what intervals of the real axis $I$ the series $\sum_{n=1}^\infty f_n$ ...
2
votes
1answer
37 views

Show that $\displaystyle\int _{(0,1) \times (0,1)} \frac {xy } {(x^2+y^2)^2 } d(\mu \times \lambda )$ does not exist?

I want to show that $$\int _{(0,1)\times (0,1) } \frac {xy } {(x^2+y^2)^2 } d(\mu \times \lambda )$$ doesn't exist, but don't know how to do it. First thing would be to try to show that the the ...
2
votes
0answers
36 views

Lebesgue-Stieltjes integral w.r.t. measure defined by absoluting continuous $F$

I know that if $F:[a,b]\to\mathbb{R}$ is a non-decreasing absolutely continuous function then$$\int_a^b f(x)dF(x)=\int_a^b f(x)F'(x)d\mu$$where the first integral is the Lebesgue-Stieltjes integral ...
1
vote
1answer
33 views

Convergence in Measure Implies Integrable

Let $f_n$ be a sequence of measurable functions which converge in measure to a function $f$. My first question is, is $f$ itself necessarily measurable? Now suppose that $|f_n| \leq |g|$ for some ...
0
votes
5answers
48 views

Use Dominated Convergence Theorem to prove that $\lim_{n\rightarrow \infty}\int_0^\pi\sqrt{\frac{t}{n}}\sin(\sqrt{\frac{n}{t}})\,dt$

Here it is function, $$\lim_{n\rightarrow \infty}\int\limits_{0}^{\pi}\sqrt{\frac{t}{n}}\sin\sqrt{\frac{n}{t}}\;dt$$ but I read about Dominated convergence theorem and I don't know how to implement ...
1
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1answer
36 views

Proving that a mass distribution has positive Lebesgue measure

I am confused in this proof about how we obtain $\int f(u) \, d\mu(u) = \int f(u)g(u) \, d\mu(u)$ and how Plancherels theorem has been applied in $(6.6)$. Furthermore, I cannot understand how if ...
1
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1answer
47 views

Sequence of simple functions defining Lebesgue integral chosen monotonic

I often find it stated in Kolmogorov-Fomin's (for ex. here in the proof of ex. 2) that the sequence $\{f_n\}$ of summable simple functions uniformly converging to $f$, that are used to define the ...
2
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0answers
17 views

Risk in density estimation: grasping the definition

When generalizing estimators to an entire function what is the space in which we perform the integral to obtain the expected value (with respect to this function)? For example, when estimating ...
2
votes
0answers
55 views

$\sup\limits_{\phi} \int_{[0,1] } \log \phi = \int_{[0,1]} \log f$

Let's say we have a measurable function $f:[0,1] \rightarrow (0, \infty)$. Approximate $f$ from below by a simple function $\phi$, with $\phi(x) > 0$ for all $x$. Then $$\int f = ...
0
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1answer
36 views

Generalised derivative and derivative of functions of bounded variation

Let $f:\mathbb{R}\to\mathbb{C}$ be a function Lebesgue-integrable on any finite interval and let $K$ be the space of infinitely differentiable equal to 0 outside a given finite interval. Be the ...
0
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0answers
35 views

Showing a sequence of integrals converges.

I'm having trouble with this problem - I don't even know how to begin. Thoughts? Solutions with explanation? Please help! Let $f$ be a bounded continuous function on $\mathbb{R}$. Prove that $$ ...
1
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1answer
26 views

Proving an equality involving a Lebesgue integrable function on R.

I'm having trouble with the following equality. I'm not even sure how to begin. Please help. Let $f$ be a real-valued, Lebesgue integrable function on $\mathbb{R}$. Prove that $$ \lim_{t \to 0} ...
1
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1answer
24 views

A sequence of Lebesgue integrable functions.

My friend and I came upon a problem in Real Analysis. It called for a sequence of Lebesgue integrable functions $(f_n)$ converging everywhere to a Lebesgue integrable function $f$ such that $$ ...
1
vote
2answers
37 views

Show inclusion of $L^p$ spaces in a space of finite measure

Let $1 \leq p_1 \leq p_2 \leq +\infty$. Show that in a space of finite measure we have that $L^{p_2} \subset L^{p_1}$. Could you give me some hints what I could do??
0
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0answers
27 views

borel sigma algebra and lebesgue measure and integral over lebesgue measure

I have a question which is writen below, would you plz help me with those hints to can to slove it: let A be borel $\sigma$ algebra on$ \Omega =[0,\infty)$ and let $\mu$ be a $\sigma$ -finite ...
0
votes
0answers
41 views

How can I prove that this function doesn't have a second weak derivative?

I'm trying to determine what weak derivatives the function $$ f(x)=\begin{cases} x&\mbox{if }0<x<1,\\ 1&\mbox{if }1\leq x<2, \end{cases} $$ has. I already managed to prove that it ...
1
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1answer
41 views

If $f$ is in $L^p$, prove that $\lim_{\lambda \to 0} \lambda ^p \omega(\lambda) = 0$

Suppose that $E \subseteq \mathbb{R}$ is measurable and that the measurable functions $f: E \to \mathbb{R}$ satisfies $\int_E |f| ^p < \infty$. If $\omega$ is the distribution function of $f$, ...
3
votes
3answers
88 views

How to prove that $L^p [0,1]$ isn't induced by an inner product? for $p\neq 2$

I'd like to know how could i prove that $L^p [0,1]$ isn't induced by an inner product? (For $p\neq 2$, including $p=\inf$). It is clear to me that i would need to find two functions $f$, $g$ in $L^p$ ...
0
votes
1answer
15 views

Derivative of Lebesgue integral function at the endpoints

Let $f$ be a non-decreasing Lebesgue-integrable real function on $[a,b]$. I read in Kolmogorov-Fomin's Элементы теории функций и функционального анализа (p. 340 here) that $$\lim_{h\to ...
0
votes
1answer
44 views

Measurability of derivative of Lebesgue integral function

I read in Kolmogorov-Fomin's Элементы теории функций и функционального анализа that if $f:[a,b]\to\mathbb{R}$ is a Lebesgue-summable function on its domain then the derivative $\Phi'$ of the integral ...
1
vote
1answer
30 views

Definition of Summable

I am studying some material on real analysis. The word "summable" troubled me. I searched online. It looks like summable means having a Lebesgue integral. But Lebesgue integral is not a familiar ...
4
votes
1answer
31 views

Is there a simpler approach to this application of Dominated Convergence?

For a measure theory class, I'm trying to evaluate: $$\lim_{n\to\infty}\int^\infty_1\frac 1 {nx} e^{-x/n}\ \text d\lambda$$ Obviously I want to try and move the limit through the integral and ...
3
votes
1answer
31 views

Proving functions are in $L_1(\mu)$.

Let $(\mathcal{X}, \mathcal{A}, \mu)$ be a measure space. Take $f,g \in L^1(\mu)$. Prove that $\sqrt{f^2+g^2}$ and $\sqrt{\vert fg\vert}$ are in $L^1(\mu)$. First, I prove that $h = ...
0
votes
1answer
40 views

Measurability of $\{(x,y): x\in M,0\leq y\leq f(x)\}$

Let $(X,\mathfrak{S}_x,\mu_x)$ be a measure space endowed with the $\sigma$-additive and complete measure $\mu_x$ defined on the $\sigma$-algebra $\mathfrak{S}_x$, let $\mu_y$ be the linear Lebesgue ...
0
votes
1answer
61 views

Gaussian distributions - a question about convergence

Let $\mu_n$ be Gaussian distributions with mean $0$ and standard deviation $1/n$ and $f$ a function. It may be true that if $\underset{\mathbb{R}}{\int} f \mu_n dx \rightarrow ...
0
votes
1answer
27 views

Proof completeness of $L^p$

I'd like to check if I understood the proof that $L^p$ is complete ($1 \le p <+\infty$). I have to use the following fact: in a metric space, if a Cauchy sequence has a convergent subsequence then ...
1
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0answers
26 views

approximation simple functions with finite support

Let $f$ be a nonnegative measurable function. I want to prove that there is an increasing sequence of nonnegative simple functions each of which vanishes outside a set of finite measure such that ...
1
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1answer
28 views

Union of $x$-sections measurable?

I know that the $y$-section $A_x$ of a $\mu_x\otimes \mu_y$-measurable set $A$, where $\mu_x\otimes \mu_y$ is the Lebesgue extension of the product measure $\mu_x\times \mu_y$ (both measures being ...
0
votes
1answer
46 views

Calculating a limit of integrals

I am having a problem with the following exercise: Show that for every bounded borelian function $\varphi : \mathbb{R} \rightarrow \mathbb{R}$, $\underset{n}{lim} \frac{n}{\sqrt{2\pi}} ...
0
votes
1answer
22 views

Lebesgue integral involving distance function

Suppose $F$ is a closed set in $\mathbb{R}$, whose complement has finite measure, and let $\delta(x)$ denote the distance from $x$ to $F$, that is $$\delta(x)=d(x,F)=\inf\{ | x -y | : y \in F ...
0
votes
2answers
29 views

Is $\frac{\vert x+y\vert}{1+\vert x+y\vert}\leq\frac{\vert x\vert}{1+\vert x\vert}+\frac{\vert y\vert}{1+\vert y\vert}$ true?

Is $\frac{\vert x+y\vert}{1+\vert x+y\vert}\leq\frac{\vert x\vert}{1+\vert x\vert}+\frac{\vert y\vert}{1+\vert y\vert}$ true for all $x,y\in\mathbb{R}$? If not, how can I prove that $\int\frac{\vert ...
0
votes
0answers
41 views

Additivity of Lebesgue integral w.r.t. sets on non-finite domain

I know that for any Lebesgue integrable function $f:X\to\mathbb{C}$, or $f:X\to\mathbb{R}$, where $X$ is a set of finite measure such that $X=\bigcup_n A_n$, $\forall i\ne j\quad A_i\cap ...
0
votes
1answer
27 views

Integral of a product with any continuous function which has integral 0 is equal to 0

Let $g:[0,1]\to\mathbb{R}$ be bounded and measurable. For every continuous function $f$ with $\int_0^1f(x)dx=0$, $\int_0^1f(x)g(x)dx=0$ holds. I want to prove that $g$ is a constant function on ...
2
votes
1answer
44 views

On convergence a.e and convergence measure

I have a question. First, I know that convergence in measure of a sequence of functions $f_n$ is different than convergence a.e., wich means there are sequences that converge in measure but not a.e. ...
1
vote
0answers
56 views

Prove $f(x)=x$ is Lebesgue integrable on $[0,1]$

Prove that $f(x)=x$ is Lebesgue integrable on $[0,1]$. My definition of integrable comes from Royden's Real Analysis (4th ed). So $f$ is integrable if the lower integral is equal to the upper ...
0
votes
1answer
21 views

Approximate measurable function by simple function with compact support

Let $f$ be a nonnegative Lebesgue measure function on $\mathbb{R}$, $\epsilon>0$. How can we approximate $f$ by a nonnegative simple function $s$ with compact support s.t. $s\leq f$ and ...
6
votes
2answers
117 views

Limit of Lebesgue integrable function

Let $f$ be a real valued, Lebesgue integrable function on $\mathbb{R}$. Prove that $$\lim_{t \to 0} \int_{\mathbb R} |f(x+t)-f(x)|\, dx=0.$$
1
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1answer
70 views

Monotone Convergence theorem

Give an example of a sequence of Lebesgue integrable functions $\{f_{n}\}$ converging everywhere to a Lebesgue integrable function $f$ such that $$\ \lim_{n \to \infty} \int_{-\infty}^{+\infty} f_{n} ...
0
votes
1answer
52 views

$\int_X f(x)\,d\mu\,\,$ exists iff $\,\,\int_X \lvert \,f(x)\rvert\,d\mu\,\,$ does

I know that, for a domain of finite measure $X$, provided that $f$ is measurable, each of the Lebesgue integrals$$\int_X f(x)d\mu\quad\text{ and }\quad\int_X |f(x)|d\mu$$exists if and only if the ...
1
vote
1answer
28 views

Bounding a linear functional in $L_2[0, 1]$

For each f in $L_2[0, 1]$ let $\phi(t)$ be the solution of $y' + ay = f$ that satisfies $\phi(0) = 0$, where a is a constant. Define $l: L_2[0,1] \to \mathbb{C}$ by $l(f) = \int_0^1 \phi(t) dt.$ ...
0
votes
1answer
10 views

Euality between these two L1 integrable functions?

Let $$\lim_{n\to\infty}\int |u_n v_n-uv| d \mu=0$$ I want to show that $$\lim_{n\to\infty}\int u_n v_n d \mu=\int u v d \mu$$ What I have so far: $$\lim_{n\to\infty}\int u_n v_n d \mu-\int u v d ...
1
vote
1answer
55 views

Measurability and a integral

I need to calculate $\lim_{n\rightarrow\infty}\int^{\infty}_{0}\frac{cos(\frac{x}{n})}{2^x}d\lambda(x)$ and show that the integral makes sense for every $n$. My approach so far: Let ...
1
vote
1answer
51 views

If $\mathbb{E}(X^\alpha)<\infty$ for $0<\alpha<1$ show that $\mathbb{E}(\min (X,t))$ is $o(t^{1-\alpha})$

If $X\geq 0$, and $\mathbb{E}(X^\alpha)<\infty$ for $0<\alpha<1$ show that $\mathbb{E}(\min (X,t))$ is $o(t^{1-\alpha})$. We know that that $$\mathbb{E}(\min(X,t))=\int_{X\leq ...
0
votes
1answer
59 views

If the Lebesgue integral of a strictly positive function is zero…

If the Lebesgue integral (over a set A) of a strictly positive function is zero, it means that the Lebesgue measure of A is zero? Thank you!
1
vote
1answer
45 views

Lebesgue integral and anti-derivative

For which Lebesgue measures the Lebesgue integral of a differentiable function over a Euclidean space or an orientable manifold coincides with its anti-derivative? For example, can we find the class ...
0
votes
1answer
28 views

Exponential limit on sum of probabilities guarantees the product of powers of expectations is integrable

If X, Y are random variables and there exists a constant $c>0$ so that $P(|X| \geq x) + P(|Y| \geq x) \leq e^{-cx}$ for all x > 0, then $E[X^m Y^n]$ is integrable for all nonnegative integers m, ...