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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4
votes
4answers
429 views

Proof of Riemann-Lebesgue lemma

I read a book, and this mention to the following lemma of Rieman-Lebesgue type. Lemma. Let $-\infty<a<b<\infty$ and $f(x,y):[a,b]^2\to\mathbb R$ be an integrable and nonnegative function. ...
0
votes
0answers
22 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} ...
2
votes
3answers
47 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 ...
6
votes
1answer
239 views

How to apply Borel-Cantelli Lemma?

Assume that we are given a sequence of continuous functions $f_n(x)$ on $[0,1]$. How to show the existence of a sequence $a_n$ and a set $A$ with $\mu(A^c)=0$ so that $$ \lim_{ n\to \infty} ...
2
votes
1answer
38 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 ...
2
votes
2answers
127 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 ...
6
votes
2answers
144 views

Invariance of the Lebesgue integral.

Problem Let $f\in L^1(\mathbb{R})$. Show that $\int_{\mathbb{R}}f(x)dx=\int_{\mathbb{R}}f(x-\frac{1}{x})dx$. Discussion I know the Lebesgue integral is translation invariant (as the Lebesgue measure ...
4
votes
0answers
29 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 ...
0
votes
1answer
30 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) ...
1
vote
0answers
35 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: ...
1
vote
0answers
23 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 ...
0
votes
0answers
31 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; ...
2
votes
1answer
49 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 ...
1
vote
2answers
52 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 ...
0
votes
1answer
27 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 $f\in L^+$ ...
0
votes
0answers
14 views

Reimman and Lebesgue

I am trying to prove the assertion below but getting stuck and always returning to the starting point. I am trying to Introduce the lebesgue as an integral of simple function, But having a major ...
3
votes
1answer
105 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 ...
0
votes
1answer
36 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 ...
-2
votes
0answers
40 views

Integral of a function in terms of size of sets where it's big.

Let $(X,\sigma,\mu)$ be a measure space and $f:X \rightarrow \Re$ a positive measurable function. For all $t > 0$ we define $M_{f}(t)= \{ x \varepsilon X: f(x) > t \} $ and $\varphi_{f}(t) = ...
0
votes
1answer
43 views

Needed help in Lebesgue Integral

Let $(X,\Sigma,\mu)$ be a measure space and $(C_n)_{n \geq 1}$ a sequence of measurable sets. Let $f:(X,\Sigma) \rightarrow (\Re, B(\Re))$ an integrable function such that $\int_{X} |f - ...
0
votes
1answer
16 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 $\int_{A}f dm<\infty$. ...
0
votes
1answer
29 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 ...
2
votes
0answers
22 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 ...
1
vote
0answers
39 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: ...
1
vote
1answer
25 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 ...
1
vote
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 ...
1
vote
1answer
27 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 ...
0
votes
1answer
27 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 ...
1
vote
1answer
37 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 ...
1
vote
0answers
33 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 $n\in \mathbb{N}$?
0
votes
1answer
39 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 ...
1
vote
2answers
26 views

Lebesque 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 ...
1
vote
1answer
23 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 ...
0
votes
1answer
25 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 ...
1
vote
0answers
46 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 ...
1
vote
1answer
27 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 ...
0
votes
1answer
15 views

condition for which we have an integrable function

Let $\Omega=[-L,L] \subset \mathbb{R}$, and let $n=\dfrac{u_x}{|u_x|}$. Now my question is what are the conditions on $\gamma(n)$ and $u_x$ so that we have $$\gamma^2(n) u_x \in L^1$$ i.e. ...
9
votes
0answers
126 views

Topology of convergence in measure

Currently I am doing some measure theory(on $X=[0,1]$ with the Borel-Sigma algebra and the Lebesgue measure) and I am looking at sets$A \subset L^p$, such that for all $q \in (0,p)$, the topologies ...
2
votes
2answers
41 views

Lebesgue integrability and measurable functions

Let $f$ be a nonnegative function on the reals. What does the (Lebesgue) measurability of $f$ have to do with the (Lebesgue) integrability of $\int f$? I've spent some time studying the definition at ...
1
vote
1answer
37 views

If $u \in L^2(\Omega)$, then $\text{sign}u \in L^2(\Omega)$?

If $\Omega$ is a bounded domain and $u$ is in $L^2$, why is $\text{sign}(u) \in L^2?$ I am only stuck with the measurabilituy part. the integral is obviously finite on a bounded domain.
2
votes
0answers
25 views

Proving this is Lebesgue integrable using radial functions

Show that $f:\Bbb R^n\to\Bbb R$, given by: $$ f(x) = \begin{cases}\sin\left(\frac{1}{\|x\|}\right)\|x\|^{-n-\arctan(\|x\|-1)} & x\not=0 \\ 0 & x=0 \\ \end{cases}$$ is Lebesgue ...
0
votes
1answer
58 views

Lebesgue integrable function without compact support.

Suppose $f \in L^1(\mathbb{R}, m)$, where $m$ is Lebesgue measure. By definition we have $$\int_{\mathbb{R}} f dm < \infty$$ Does $f$ have compact support? This makes sense, but I don't know if ...
0
votes
0answers
18 views

How can I calculate the following Lebesgue integral $\int_{-\infty}^{\infty}||x|^{-0.35}-|x-1|^{-0.35}|^{1.8} d\lambda(x)$

How can I calculate the following Lebesgue integral? Is it convergent? $\int_{-\infty}^{\infty}||x|^{-0.35}-|x-1|^{-0.35}|^{1.8} d\lambda(x)$ where $\lambda$ is the Lebesgue measure.
0
votes
1answer
35 views

Lebesgue integration: Existence of double integral, but not Lebesgue integrable.

I am trying to determine whether or not $f(x,y) = \dfrac{\sin(x)\sin(y)}{x^2+y^2}$ is integrable on $E = \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \times \left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ ...
1
vote
1answer
42 views

Lebesgue Integration: Double Integral (Fubini)

I'm trying to determine whether or not $f$ is integrable on $E$, where $f(x,y) = e^{-xy}$ and $E = \{(x,y) : 0 < x < y < x+x^2\}$ Ok, so $f$ is continuous and non-negative on $E$ so it is ...
3
votes
1answer
34 views

Integration by parts in Bochner Lebesgue spaces.

Does there exist an analogous of integration by parts for expressions such as: $$\int_0^T {\langle u(t),v(t) \rangle }\, \mathrm{d}t,$$ where $u,v\in L^2([0,T];H)$, for some Hilbert space $H$? If so, ...
0
votes
1answer
24 views

lebesgue integral

let $f\ge 0$ be a measurable function s.t. $\int_R fdm=\infty$, show that for any M>0 there is a real measurable function g, and $0\le g \le f$ and the following hold: $\int_R g dm \ge M$ and g is ...
0
votes
1answer
32 views

lebesgue measure and integral

assume that f is a non-negative real-velaued measurable function, and $\int_R f(x)dm<\infty$ (lebesgue measure) show for any real number a, a is not 0, $\int_R f(ax)dm=\frac{1}{|a|} \int_R f(x)dm$
3
votes
1answer
33 views

Lebesgue integral over “bad” measurable sets

Let $\Omega \subset \mathbb{R}^n$ ($n \geq 1$) be a bounded open domain and $f \in L^\infty(\Omega)$ possibly changes the sign. Assume that the set $$ \Omega^+ := \{x \in \Omega: f(x) > 0 \} $$ has ...
2
votes
1answer
30 views

Understanding the proof of completeness of $L^1$.

I'm reading the proof of completeness of $L^1 (X, \mathscr{M}, \mu)$, and I would like to clear up some confusion To prove $L^1$ is complete it suffices to show that every Cauchy sequence $(f_n)$ has ...