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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2
votes
2answers
61 views

Why this doesn't contradict Monotone and Dominated Convergence Theorem? [closed]

$$\lim_{n \to \infty} \int_0^\infty f_n(x)dx \ne \int_0^\infty \lim_{n \to \infty} f_n(x)dx$$ where : $f_n=ne^{-nx}$ for all $x \in [0,\infty)$ $n \in \mathbb{N}$ Can somebody help me with ...
1
vote
1answer
49 views

Question about inequality relating infinity norm and Lebesgue integral

I have a question about the following question: Let $f:\mathbb{R} \to \mathbb{R}$ be a measurable function. We know there exists a constant $K$ such that for every bounded continuous function ...
0
votes
0answers
24 views

Fourier transform isometry

I want to show that for a sufficiently fast decaying function we have $$\pi \int_{\mathbb{R}^3} \frac{f(x)}{|x|} dx = \int_{\mathbb{R}^3} \frac{\widehat{f(x)}}{|x|^2} dx. $$ Does anybody see where ...
2
votes
1answer
38 views

Attempting to use dominated convergence theorem

I am trying to prove the following: Let $(X,\Sigma, \mu)$ be a measure space and let $f:X\rightarrow [0,\infty)$ be an integrable function. Then for any real number $\alpha>1$, $$\int_X n ...
0
votes
0answers
33 views

Find a function $b$ such that the operator $\frac{d}{dx}+b(x)$ is symmetric with the weight $x^2$

Find the value of $b(x) \in \mathbb{C}, x\in \mathbb{R}$, so that $$Â=(Â^{*})^{t}$$ with $$Â=i\frac{d}{dx}+b(x)$$ Here, $(f|g)$ is defined by $$ \int_{-\infty}^{\infty} x^{2}f^{*}(x)g(x)dx $$ I ...
1
vote
0answers
38 views

Evaluate Integral of $\int_0^{\frac{\pi 2^n}{2}}\sin^{2+d}(u)\prod_{i=1}^{n}\cos^{2-d}(u/2^i)du$

Integral of $$\int_0^{\frac{\pi 2^n}{2}}\sin^{2+d}(u)\prod_{i=1}^{n}\cos^{2-d}(u/2^i)du$$ I've tried a number of ways to re-write this in a way that makes sense, but no luck thus far. The integrand ...
0
votes
0answers
25 views

Lebesgue integral problem, approximation by continuous function? [duplicate]

I have the following problem, it seems quite simple, but I think the simplicity makes it harder to get started. Could you guys tell me where to star> I was thinking this might be application of ...
3
votes
2answers
68 views

Hint for Lebesgue theory/functional analysis type of problem

I am trying to solve the following problem, but I am not too familiar with functional analysis. Could you guys tell me where I should start? Thanks! Let $f \in L^1(\mathbb{R})$ and define $$f_n(x) = ...
0
votes
1answer
30 views

For which $\alpha, \beta \in \mathbb{R}$ is $\frac{1}{x^\alpha (\log(x))^\beta} \in L^p((1,\infty)$ where $1\leq p \leq \infty$

I'm having a really hard time finding a starting point here. I think that it won't work for any $\alpha, \beta >0$ but I am not even sure about that.
3
votes
1answer
49 views

Prove an equality ($L^P$ spaces)

Prove the equality $$\int |f(x)|^p dx=\int_0^\infty pt^{p-1}m(\left\lbrace x:|f(x)|\geq t\right\rbrace)dt$$ for $p\geq 1$. My first idea was to try to prove this via induction. For the case $p=1$, ...
4
votes
3answers
110 views

Integral equal $0$ for all $x$ implies $f=0$ a.e.

Let $f\in L^2[0,1]$. $$\int_0^x f(t)dt =0 \quad \forall\, x\in(0,1) \Longrightarrow f=0 \text{ a.e}$$ What is the easiest proof?
0
votes
1answer
23 views

Lebesgue-integrable function particular characteristic

Let $\phi:\Bbb R\to\Bbb R$ be a function given by: $$\phi(x)= \left\{ \begin{array}{ll} x^{-1/2} & x\in (0,1) \\ 0 & x\notin (0,1) \\ \end{array} \right.$$ Let $\sum_{k=1}^\infty ...
0
votes
1answer
15 views

Integration w.r.t nondegenerate Gaussian probability measure on $X$ with mean $0$

Suppose that $X$ is a Banach space. Denote $\gamma$ as a nondegenerate Gaussian probability measure on $X$ with mean $0$. Question: Is it true that $$\int_X{d\gamma(t)}=0?$$ Or we have ...
0
votes
0answers
15 views

Existence of a function that is not Lebesgue-integrable for any $q\in (0,\infty)$

So the proof is to show a continuous function $f:(0,a)\to \Bbb R\; (a>0)\;$ such that: $$f\notin L_q(\lambda)\quad \forall q\in (0,\infty)$$ So, lets consider $\; f(x)=e^{\frac{1}{x}}$, then since ...
0
votes
1answer
43 views

If $f ∈ L^1((0, 1))$ is non-negative and $\int_0^1 f=1$, then $\int_0^1 f^{-1}\ge 1$

Assume that $f ∈ L^1((0, 1))$ is a non-negative real valued function satisfying $$\int_{0}^{1}f(x) dx = 1 $$ Show that $$\int_{0}^{1} \frac{1}{f(x)} dx≥ 1 $$ if we can show $ \frac{1}{f(x)}$ > 1 a.e. ...
0
votes
0answers
31 views

For a Lebesgue integrable function $f$, the map $t\mapsto\int \chi_{A+t}f(x) dx $ is continuous

Suppose that $ f\in L^1(\mathbb{R})$ and $A $ is Borel subset of $R$. Show that the mapping $t$ to $\int \chi_{A+t}f(x) dx $ is continuous from $\mathbb{R}$ to $\mathbb{R}$. I try first by taking ...
1
vote
2answers
46 views

Prove that $\|f\|_p \leq \liminf \|f_n\|_p$ under weak convergence

Let $1<p<\infty$ and $q$ its conjugate. Given a sequence $(f_k)_{k \in \mathbb N}$ and $f$ in $L^p(\mathbb R^d)$, I am trying to show that if for all $g \in L^q(\mathbb R^d)$, $$\lim_{k \to ...
6
votes
1answer
76 views

Limit of $\int_E f(nx) dx$ for a $1$-periodic function $f$ on $[0,2\pi]$

Let $E$ be a measurable subset of $[0, 2\pi]$. Assume that $f \in C(\mathbb R)$ is $1$-periodic, i.e. $f(x + 1) = f(x)$. Compute $$\lim_{n\to\infty} \int_{E} f(nx) dx$$. Since $f$ is continuous on ...
0
votes
2answers
62 views

Is $f(x,t) = \sin (x) e^{-tx}$ in $L^1([0, \infty) \times [0, \infty))$?

I have the integral $\displaystyle\int_0^\infty \int_0^\infty \sin (x) e^{-tx} \ dt \ dx.$ I want to switch the order of integration using Fubini's Theorem, which requires that the integrand be ...
3
votes
1answer
60 views

Show $f$ is integrable

Let $f$ be such that $\int_0^{\infty} |f(s)|e^s ds< \infty.$ Now, I want to argue that for $x,y$ sufficiently large and $\lambda < 1$ fixed we have that $$\int_0^{\infty} \int_x^y e^{\lambda ...
1
vote
0answers
38 views

Problem related to $L^p$ and distribution function

I am trying to solve the following problem: Let $\lambda_f(t) := \mu(\{x \in X: |f(x)| > t\})$ Prove that $f \in L^p(\mathbb R^d)$ for $0<p<\infty$ if and only if ...
1
vote
1answer
59 views

Convergence in L^p and convergence in norm

I am trying to show these two statements: Let $(\mathbb R^n,\mathcal M,m)$ where $m$ is the Lebesgue measure and $\mathcal M$ are the Lebesgue measurable sets, and let $(f_n)_{n \in \mathbb N}$ and ...
1
vote
0answers
43 views

Show that $v(t) = \int_0^1 e^{\sqrt{x^2+t^2}} d\lambda$ is continuous.

I'm trying to show that the function $v(t) = \int_0^1 e^{\sqrt{x^2+t^2}} d\lambda$ (with $\lambda$ being the Lebesgue measure) is continuous for all $t \in \mathbb{R}$. I've however hit a little ...
0
votes
1answer
42 views

Proving that $\int 0 d\mu=0$

While looking for a clean proof of $\int 0 d\mu=0$, I encountered a difficulty. Consider $0$ as a function between $(X,\Sigma, \mu)$ and $({\mathbb R}, \mathcal B)$. Since $0=1_\emptyset+0\cdot ...
1
vote
1answer
41 views

Positive function with finite integral is finite almost surely

Let $\displaystyle f:(X,\Sigma, \mu)\to (\bar{\mathbb R_+}, \mathcal B)$ be a measurable non-negative function that may assume the value $+\infty$. Suppose that $\int f d\mu$ is finite. Prove ...
2
votes
1answer
51 views

Prove that $\int$fd$\mu$ $\leq$ $\int$gd$\mu$

Let f, g $\in \mathcal{L}^{+}$ such that f(x) $\leq$ g(x) for all x $\in$ X. Prove that $\int$fd$\mu$ $\leq$$\int$gd$\mu$. I think I figured out the proof and I am curious as to whether or not I'm on ...
2
votes
1answer
61 views

Can $L^\infty$ function be redefined so that every point is a Lebesgue point?

Consider a reasonable subset $\Omega$ of an Euclidian space with a scalar-valued $f \in L^\infty(\Omega)$. (I'll define reasonability later.) It is well-known that almost every point of $\Omega$ is a ...
1
vote
2answers
42 views

How to write down, think about, and evaluate a simple Lebesgue integral

I sometimes come across integrals of the following form, which I think are related to the Lebesgue integral: $$\int_A f(x)d\mu(x).$$ What does the $d\mu(x)$ mean? I think this means integrate with ...
0
votes
1answer
19 views

Non-existence of convex neighbourhoods in $L^p(0,1)$

Problem Let $0<p<1$, show that the neighbourhoods $\{f \in L^p(0,1):||f||_p<\epsilon\}$ of the zero function are not convex. I am pretty stuck with this problem. If I've understood the ...
3
votes
3answers
71 views

Meaning of $\int_E {f(x) \mu(dx)}?$

Suppose $f$ is a measurable real-valued function defined on a measure space $(E, X , \mu)$. What is the meaning of the RHS of the following integral $$\int_E{f d\mu} = \int_E {f(x) \mu(dx)}?$$ I ...
3
votes
2answers
80 views

Integral of $\int_0^\infty \frac{\sin^4(u)}{u^{k}}du$ where $k\in(1,3)$

My task is to Evaluate $$\int_0^\infty \frac{\sin^4(u)}{u^{k}}\,du$$ where $k\in(1,3).$ I've tried a few things, but nothing seems to be working. Any help?
0
votes
0answers
50 views

The Laplace transform of an integrable function is differentiable

let $f\in L^1(0,\infty)$. For x>0, define $g(x)=\int_{0}^{\infty} f(t) e^{-tx} dt$. Prove that $f$ is differentiable for $ x>0$ and with derivative $g'(x) = \int_{0}^{\infty} -tf(t) e^{-tx} dt$. ...
1
vote
0answers
40 views

If $\{f_n\}$ is a Cauchy sequence in $(L^1,\|\cdot \|_{L^1})$ does it imply $f_n\to f$ for a certain $f\in L^1$?

In the book "Real analysis" by Stein, the theorem 2.2 (Reisz-Fischer) says: The vector space $L^1$ is complete in its metric. To me, it means that if $(f_n)$ is a Cauchy sequence for $\|\cdot ...
1
vote
1answer
21 views

bounded convergence, is the fact that $m(E)$ finite necessary?

The bounded convergence theorem says that if $\{f_n\}$ is such that $f_n$ are measurable, bounded by $M$ for all $n$, supported on a set $E$ of finite measure and $f_n(x)\to f(x)$ a.e. as ...
0
votes
1answer
45 views

Lebesgue Integrability Criteria

I want to prove T(x)=(sin x)^(k) is Lebesgue integrable on [o,pi] how can i define upper and lower bounde to show? and solove? please tell for all value of k, is this Lebesgue integrable? for odd or ...
0
votes
2answers
39 views

A nice property of L1 function in [0,1] .

Let f is in $L^1$$([0,1],m)$,$m$ is a Lebesgue measure and suppose that $f(x)>0$ for all $x$. Show that for any $ 0<\epsilon<1 $ there exists $\delta$ >0 so that $\int_{E} f(x)dx\ge \delta$ ...
1
vote
1answer
68 views

Let $E$ be a Lebesgue measurable subset of $\mathbb{R}$ such that $m(E\cap(E+t)) = 0$ for all $t \neq 0$. Then prove that $m(E)=0$.

Let $E$ be a Lebesgue measurable subset of $\mathbb{R}$ such that $m(E\cap(E+t)) = 0$ for all $t \neq 0$. Then prove that $m(E)=0$. I think that the function $f(t) = m(E\cap(E+t))$ is continuous but ...
0
votes
0answers
40 views

Prove that $\lim_{x\to\infty} f(x) = 0$ if $\lim_{x\to\infty}( f(x)+ \int_{0}^{x} f(x) dx)$ exists [duplicate]

Let f be a real valued continuous function on $[0,\infty)$ such that $\lim_{x\to\infty}( f(x)+ \int_{0}^{x} f(x) dx)$ exists. Prove that $\lim_{x\to\infty} f(x) = 0 $. If f is non negative function ...
2
votes
3answers
47 views

Royden's Lebesgue Integration 4.4 #34

Let $f$ be a nonegative measurable function on $\mathbb{R}$. Show that \begin{equation} \boxed{\lim_{n \to \infty} \int_{-n}^{n} f=\int_{\mathbb{R}} f} \end{equation} Of course $$\forall n, ...
1
vote
0answers
21 views

Lebesgue integrable function composed with Continuously Differentiable Bijection

Assume $f:\mathbb{R}\to\mathbb{R}$ is a Lebesgue integrable function and let $\phi:\mathbb{R}\to\mathbb{R}$ be a continuously differentiable bijection. Show that: $$\int_\mathbb{R} f\circ ...
2
votes
1answer
34 views

Help in understanding a property of Lebesgue measure and linear applications.

I've been trying to understand why this relation is true and can't find the right way to attack it. Let $\Phi$ be a linear map from $R^d$ to $R^d$ and $\lambda$ the Lebesgue measure in $R^d$, then ...
1
vote
0answers
28 views

Lower bound for maximal function which yields maximal function is not integrable [duplicate]

Let $f : \mathbb{R}^n \rightarrow \mathbb{C}$ be Lebesgue integrable and $f$ is not identically zero (i.e., $m(\{x \mid f(x) \neq 0\}) >0$). Then there exist constant $C, R > 0$ such that ...
1
vote
0answers
93 views

Chain rule for Radon-Nikodym derivative (without a.e.)

Assume that $\mu, \nu, \rho$ are $\sigma$-finite positive measures. If $\nu << \mu, \rho << \nu$, then $\rho << \mu$ and $$\frac{d\rho}{d\mu} = \frac{d\rho}{d\nu}\frac{d\nu}{d\mu}.$$ ...
0
votes
0answers
17 views

Lebesgue decomposition

Define $$\alpha(x) = x+1 \ \mbox{if} \ x \geq 0 \\ = x \ \mbox{if} \ x < 0.$$ Let $\nu$ be the Lebesgue-Stieljes measure corresponding to $\alpha.$ Find the Lebesgue decomposition of $\nu$ with ...
4
votes
2answers
50 views

Finding a dominating function

I have been asked to find a dominating function for the following sequence of functions: $$f_{n}(x)=\frac{x n^{3/2}}{1+n^{2}x^{2}}, x\in[0,1]$$ in order to use the dominated convergence theorem. I ...
1
vote
1answer
25 views

Evaluating the limit of $\int_{\mathbb{R}} \frac{1}{x^2+n^2} \cos(\sqrt{x^2+n^2})~d\lambda$ in the lebesgue measure.

I'm sort of stuck on evaluating the limit (as $n \to \infty$) of this Lebesgue integral, I hope someone here can help me out. The integral is: $I_n=\int_{\mathbb{R}} \frac{1}{x^2+n^2} ...
4
votes
0answers
61 views

Fubini–Tonelli theorem

By Fubini-Tonelli theorem, if one of $\int(\int |f(x,y)| dy)dx$, $\int(\int |f(x,y)| dx)dy$, $\int\int |f(x,y)| dxdy$ is finite , then $\int(\int f(x,y) dy)dx = \int(\int f(x,y) dx)dy=\int \int ...
2
votes
2answers
67 views

Integral with smaller sigma algebra

Here's an exercise: Let $(X,M,\mu)$ be a measure space with $\mu(X)<\infty$. Let $N\subseteq M$ be a $\sigma$-algebra. If $f\geq 0$ is $M$-measurable and $\mu$-integrable, then there exists some ...
1
vote
1answer
48 views

How to show continuously differentiable.

Let $$f(x)=\int_{0}^{\infty} e^{-xt} t^x\ \mathsf dt$$ for $x>0$. Show that $f$ is well defined and continuously differentiable on $(0, \infty)$ and compute its derivatives. My confusion is ...
2
votes
2answers
59 views

Primitive of an $L^1$ function is continuous

The primitive of a continuous function on a compact interval is continuous via the Fundamental Theorem of Calculus. Let $I \subset \mathbb{R}$ be open and let $u': \overline{I} \mapsto \mathbb{R}$ be ...