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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59 views

Approximating simple summable function in measure space with countable base

Let $f:X\to \mathbb{Q}+i\mathbb{Q}\subset\mathbb{C}$, $f\in L_1(X,\mu)$ be a Lebesgue-summable function taking only finitely many values $y_1,\ldots,y_n\in \mathbb{Q}+i\mathbb{Q}$ on the sets ...
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1answer
62 views

$|f+g|^p$ Lebesgue-summable if $|f|^p$ and $|g|^p$ are

I read that the Minkowski integral inequality, which I knew for Riemann integrals on $[a,b]$, holds for Lebesgue integrals in the following form:$$\forall p\geq 1\quad\quad\Bigg(\int_X |f+g|^p ...
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1answer
83 views

Coincidence of functions defining Riemann-Stieltjes integral

I read in Kolmogorov-Fomin's Элементы теории функций и функционального анализа that (p. 372 here), if the Riemann-Stieltjes integrals $$\int_a^b f(x) d\Phi_1(x)\quad\text{ and }\quad\int_a^b f(x) ...
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0answers
78 views

Applications of the Fubini-Tonelli Theorems

We are working with Lebesgue integrals right now in analysis, and in particular just proved the Fubini and Tonelli theorems. I have some questions on our latest problem set: Let $E$ be a measurable ...
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2answers
100 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 ...
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2answers
161 views

Mollifiers: Approximation

Problem Given a mollifier: $\varphi\in\mathcal{L}(\mathbb{R})$ Then it acts as an approximate identity: $$f\in\mathcal{C}(\mathbb{R}):\quad\int_{-\infty}^\infty n\varphi(nx)f(x)dx\to ...
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1answer
58 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 ...
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0answers
25 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 $D$. ...
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19 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 $\sum_{n=1}^\infty f_n$ ...
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1answer
40 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
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0answers
71 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 ...
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1answer
37 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 ...
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5answers
65 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 ...
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1answer
52 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 ...
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1answer
61 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 ...
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0answers
23 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 ...
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0answers
59 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 = ...
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1answer
110 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 ...
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0answers
37 views

Showing a sequence of integrals converges. [duplicate]

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 $$ ...
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1answer
31 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} ...
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1answer
37 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 $$ ...
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2answers
45 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??
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0answers
55 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 ...
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1answer
44 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 $f$, ...
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3answers
98 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 $L^p$ ...
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1answer
21 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 ...
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1answer
83 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 ...
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1answer
39 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 ...
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1answer
37 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
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1answer
41 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 = ...
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1answer
51 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 ...
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1answer
64 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 ...
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1answer
37 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 ...
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0answers
36 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 ...
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1answer
35 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 ...
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1answer
51 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}} ...
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1answer
26 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 ...
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2answers
31 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 ...
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0answers
51 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 ...
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1answer
39 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 ...
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1answer
66 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. ...
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0answers
80 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 ...
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1answer
41 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
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2answers
140 views

Limit of Lebesgue integrable function

Let $f$ be a real valued, Lebesgue integrable function on $\mathbb{R}$. Prove that $$\lim_{t \to 0} \int_{\mathbb R} |f(x+t)-f(x)|\, dx=0.$$
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1answer
80 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} ...
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1answer
56 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 ...
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1answer
29 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
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1answer
12 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
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1answer
59 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
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1answer
53 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 ...