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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Square Integrable Functions Formula

The book reads as follows: "Let $a(x)$ and $b(x)$ be square integrable functions defined on [$a , b$]. First we note that it follows from the elementary inequality $|ab| \le 1/2 (a^2 + b^2)$ ...
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2answers
27 views

Integral of a measurable function

I do not know what should i keep as title for this question... Question goes like this.. Let $f:\mathbb{R}\rightarrow [0,\infty)$ be a measurable function. If $\int_{-\infty}^{\infty}f(x)dx=1$ prove ...
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39 views

In which way Lebesgue Integral integrates over values?

A special tutorial for full dummies says In order to distinguish between the Lebesgue and Reimann integrals consider the values that the function f can take to be on the x-axis (called the ...
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1answer
25 views

Measurablity of functions defined over sections of product measures

I have to solve the following exercise but I am unable to proceed. Could you please give me some hints to how to solve it? Let $(\Omega_1, \mathcal{F}_1)$ and $(\Omega_2, \mathcal{F}_2)$ be ...
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1answer
35 views

Prove the following integral is asymptotically zero

I have to solve the following exercise. I would appreciate to get a hint for it. Suppose $(\Omega, \mathcal{F}, \mu)$ be a measure space. Let $f$ be an integrable function. Show ...
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1answer
31 views

Local Riesz Potential estimate in terms of Maximal Function

For $f \in L^1_{\text{loc}}(\mathbb R^n)$, and fixed $R > 0$ we defined the local Riesz potential by $$I(x) = \int_{B(x,R)} \frac{f(y)}{\lvert x-y \rvert^{n-1}} d\lambda (y), \hspace{1cm} x \in ...
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+50

Simple random walk on $\mathbb Z^d$ and its generator

I'm still trying to figure out definitions and properties of random walks on $\mathbb Z^d$. My goal is to work up to understanding some large deviation principles for the local times of such random ...
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1answer
36 views

Continuity of $F(x)=\int_{(-\infty,x]}fd\lambda$

For a homework assignment I was told to prove that given $f\in L^1(\mathbb R)$, the following function is continuous $$F(x)=\int_{(-\infty,x]}fd\lambda.$$ I thought to use DCT and show sequential ...
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2answers
33 views

Why is Monotone Convergence Theorem restricted to a nonnegative function sequence?

Monotone Convergence Theorem for general measure: Let $(X,\Sigma,\mu)$ be a measure space. Let $f_1, f_2, ...$ be a pointwise non-decreasing sequence of $[0, \infty]$-valued ...
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37 views

Show $\frac{y-x}{(2-x-y)^3}$ is not integrable on $[0,1]\times[0,1]$, not invoking Fubini's theorem.

The double integral $$I = \int_{[0,1]\times[0,1]}\frac{y-x}{(2-x-y)^3} dxdy$$ does not have a finite value. The two iterated integrals have different values (Counterexample to Fubini?). Then Fubini's ...
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Lebesgue versus Riemann integrable

Can a Lebesgue measurable function be modified on a set of first category so as become continuous except on a set of Lebesgue measure zero? OR Can a Baire-measurable function be modified on a set of ...
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32 views

Could fast or irregular oscillations make Lebesgue integral fail?

Let's consider real measurable functions defined in a bounded interval. As long as a function is bounded, oscillations at least cannot make the volume under the graph of the function infinite. But I'm ...
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2answers
34 views

$f$ has a zero integral on every measurable set. Prove $f$ is zero almost everywhere

I am trying to solve the following exercise: Let $f$ be integrable. Assume that $\int_A f d\mu = 0$ for every measurable set $A$. Prove that $f = 0$ a.e. [$\mu$]. I have the following proof but it ...
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2answers
84 views

Find $\lim_{n \rightarrow \infty}\frac{1}{n} \int_{1}^{\infty} \frac{dx}{x^2 \log{(1+ \frac{x}{n})}}$

Find: $$\lim_{n \rightarrow \infty} \frac{1}{n} \int_{1}^{\infty} \frac{dx}{x^2 \log{(1+ \frac{x}{n})}}$$ The sequence $\frac{1}{nx^2 \log{(1+ \frac{x}{n})}}=\frac{1}{x^3 \frac{\log{(1+ ...
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3answers
58 views

Find $\lim_{n \rightarrow \infty} \int_0^n (1+ \frac{x}{n})^{n+1} \exp(-2x) \, dx$

Find: $$\lim_{n \rightarrow \infty} \int_0^n \left(1+ \frac{x}{n}\right)^{n+1} \exp(-2x) \, dx$$ The sequence $\left(1+ \frac{x}{n}\right)^{n+1} \exp{(-2x)}$ converges pointwise to $\exp{(-x)}$. So ...
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1answer
21 views

Counting measure on sigma algebra power set of natural numbers .

My text book does not provide much about counting measures and integration. So I decided to setup integration on space $(N , P(N) , \mu_c ,R)$ myself imitating the construction of Lebesgue integral. ...
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1answer
21 views

Continuous map from $L^r(\Omega)$ to $L^s(\Omega)$.

The following theorem appears in the appendix of P.H. Rabinowitz monograph on Critical Point Theory: Let $\Omega \subset \mathbb R^n$ be bounded. Let $g$ be such that (i) $g \in C(\overline{\Omega} ...
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1answer
30 views

A limit of integral

How can I prove that $ \lim_{ n\to \infty} \int_{R}^{} \cos(nt)f(t)dt = 0 $ for any $f \in L_{1}(R) $? I believe that I should use a fact that cosinus is a cyclic function and divide this integral ...
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115 views

Prove $\lim_{n \rightarrow \infty} n \int_{\frac{1}{n}}^{1} \frac{\cos(x+\frac{1}{n})-\cos(x)}{x^{\frac{3}{2}}}dx$ exists.

Prove $\displaystyle\lim_{n \rightarrow \infty} n \int_{\frac{1}{n}}^{1} \frac{\cos(x+\frac{1}{n})-\cos(x)}{x^{\frac{3}{2}}}\,dx$ exists. I want to use Dominated convergence theorem to show the ...
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17 views

How to integrate with a matrix in the measure?

I've been given the following integral (actually a path integral from quantum field theory). $$ Z(a;N) = \int d^{2n}M.exp(-\frac{1}{2}tr(M^2)-\frac{a}{M}tr(M^4)) $$ where M is a square matrix of size ...
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1answer
35 views

Does $\int f(s) ds = \int g(s) ds \not =0$ imply $f(s)=g(s)$?

specifically for an improper integral, but I'm also wondering about for definite integrals. I'd guess that it's true, but I feel like there must exist different functions that integrate to the same ...
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52 views

Interchange of integral and differentiation: calculus version vs. analysis version

There are theorems that specify conditions that guarantee the interchange of differentiation and integral: $$ \frac{d}{dx}\int f(x, y) dy = \int \frac{\partial}{\partial x}f(x,y) dy $$ In calculus $f$ ...
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39 views

Measure Theory and $L^{p}$ spaces

I have the two following very simple questions regarding measure theory that I want to show: If $f \in L^{p}(X, \mathcal{M}, \mu)$ for $1 \leq p < \infty$, then $f < \infty$ $\mu$-almost ...
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35 views

Suppose there is a constant $C$ such that $\| f_n - f\|_1 \leq \frac{C}{n^2} $ for all $n \geq 1$. Show that $f_n \rightarrow f$ a.e. [duplicate]

Let $m$ be Lebesgue measure on $\mathbb R$ and let $f_n ,f \in L^1 (m)$. Suppose there is a constant $C$ such that $\| f_n - f\|_1 \leq \frac{C}{n^2} $ for all $n \geq 1$. Show that $f_n \rightarrow ...
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1answer
31 views

Let $f \in L^{1}([0,1])$ be a real valued function. Prove the following

Let $f \in L^{1}([0,1])$ be a real valued function. Prove the following $1) x^k f(x) \in L^1([0,1])$ for all $k\in \mathbb{N}$ $2) \lim_{k\rightarrow\infty}\int_{0}^{1}x^k f(x) dx = 0$ $3)$ If ...
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1answer
22 views

Prove $F$ is in $L^1$

This is an old qualifier exam question at my school Let $f \in L^{1}([0,\infty))$ and for $x\geq 0$, define $F(x) = \int_{(x,\infty)} f(t) e^{x-t} dm(t) $ Show that $F \in L^{1}([0,\infty))$ The ...
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0answers
32 views

Fubini's Theorem and expectation of random variables

I have a question regarding the application of the Fubini's Theorem to the expectation of the product of two random variables. Let $X,Y$ be two random variables defined on the probability space ...
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1answer
43 views

Prove that $\lim_n \int_{\Bbb R} \frac{\sin(n^2 x^5)}{n^2 x^4} \chi_{(0,n]} d\lambda(x) = 0$

Prove that: $$\lim_n \int_{\Bbb R} \frac{\sin(n^2 x^5)}{n^2 x^4} \chi_{(0,n]} d\lambda(x) = 0$$ I am self-learning these stuff, and I would like to check whether I did things right. Here's ...
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1answer
41 views

$\left\|f\right\|_{L^1(μ_1)}<∞$ $μ_2$-a.e.,$\left\|f\right\|_{L^1(μ_2)}<∞$ $μ_1$-a.e. $⇒$ $\left\|f\right\|_{L^1(μ_1\otimesμ_2)}<∞$

Let $(\Omega_i,\mathcal A_i,\mu_i)$ be a $\sigma$-finite measure space and $f:\Omega_1\times\Omega_2\to\mathbb R$ be measurable with respect to $\mathcal A_1\otimes\mathcal A_2$. Can we conclude, that ...
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1answer
23 views

Fubini-Tonelli theorem and absolutely Lebesuge integrable functions

As far as I know, a measurable function is Lebesgue integrable if and only it is absolutely integrable. It is simply because the definition of the integrability requires each of the positive part and ...
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27 views

Application of Leibniz rule for Lebesgue integral

Consider the real-valued random variables $X,Y$ defined on the same probability space $(\Omega, \mathcal{F}, \mathbb{P})$. Consider the function $f\colon\mathbb{R}\rightarrow [0,\infty)$. Let ...
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20 views

Different definitions of expectations: which types of integral do they involve?

Consider a random variable $X: \Omega \rightarrow \mathbb{R}$ defined on the probability space $(\Omega, \mathcal{F}, \mathbb{P})$. The probability space induced by $X$ is $(\mathbb{R}, ...
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4answers
78 views

If $ f \rightarrow c$ then prove $\frac{1}{a} \int_{[0,a]} f \rightarrow c$

Let $f$ be an extended real-valued $\mathcal{M}_{L}$-measurable function on $[0,\infty)$ such that $f$ is $\mu_L$-integrable on every finite subinterval of $[0,\infty)$, and $$ \lim_{x\rightarrow ...
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2answers
42 views

Why the following integral is Riemann integrable but not Lebesgue integrable?

I know Riemann integrable implies Lebesgue integrable, but why the following integral is Riemann integrable but not Lebesgue integrable? S=$\int_E {1\over{x-y}}dm$, where $E=[0,1]\times[0,1] $. I ...
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0answers
40 views

Existence and non-singularity of the Fisher information matrix

Consider a random vector $X$ defined on the probability space $(\Omega, \mathcal{F}, \mathbb{P})$, $X: \Omega \rightarrow \mathbb{R}^k$. Suppose $X$ has probability density $p_{\theta_0}$ with respect ...
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2answers
41 views

Lebesgue Integral of a non negative piecewise function

Consider the function over [0,1] given by $f(x)= \begin{cases} 0 & x \in \mathbb{Q}\\ x & x \notin \mathbb{Q} \end{cases}$ In order to compute the Lebesgue integral of $f$ we need to find an ...
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1answer
42 views

Book Recommendation for Measure Theory in n-Space

What's a standard book on multidimensional measure theory? I'm aware of some books on functions of several variables, but they do not discuss measure theory or Lebesgue integration in space. Thanks. ...
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59 views

Integration over finite partition of integration domain

I think the title does not reflect my problem very well. Feel free to leave a comment with a more appropriate title. Let $f \in L^1([0,1])$. How do I prove there exists a partition of $[0,1]$ into ...
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68 views

Why is linearity a requirement of a integral

I was reading Philip Protter's Stochastic Integration and Differential Equations textbook. He mentions that an operator, $I_X$, induced by $X$ should be linear to be called an integral. I have a ...
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2answers
169 views

Applications of Dominated/Monotone convergence theorem

Consider a measure $\mu$ on $(\mathbb{R}, \mathcal{B}(\mathbb{R}))$ where $\mathcal{B}(\mathbb{R})$ is the Borel $\sigma$-algebra on $\mathbb{R}$. Consider the function $f: [0,\infty)\rightarrow ...
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1answer
20 views

$f$ real-valued function that dies of in infinity but $f^p$ not integrable for any $p$.

Is there a positive continuous function on $\mathbb R$ such that $f(x) \to 0$ as $x \to \pm \infty$ but $f^p$ not integrable for any $p>0$?
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35 views

Does the function $f(x)=\frac{1}{\sqrt x}$ belong to $L^p( \mathbb N , P(\mathbb N), \mu),p=1,2,\infty?$

Does the function $f(x)=\frac{1}{\sqrt x}$ belong to $L^p( \mathbb N , P(\mathbb N), \mu),p=1,2,\infty?$ $\mathbb N$- set of natural numbers, $P(\mathbb N)$- the partitive set of natural numbers. I ...
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1answer
44 views

$\int_{\Omega} |f_n-f||f_n| \, d \mu \to 0$ if $f_n \in L^1(\Omega)$, $f_n \to f$.

Suppose $f_n \to f$ in $L^1(\Omega)$ where $\mu(\Omega)=1$. Suppose $$\int_{\Omega} |f_n| \, d\mu \leq M$$ for all $n$. Is there a way to show that the integral $$\int_{\Omega} |f_n-f||f_n| \, d ...
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1answer
28 views

Let $(X, M, \mu )$ be a space with measure. $f:X \to \mathbb R \text{ and } f\in L^1(X).$ ..

Let $(X, M, \mu )$ be a space with measure. $f:X \to \mathbb R \text{ and } f\in L^1(X).$ Prove that for all $\epsilon > 0$ that there exists $\delta > 0$ such that for $E \in M$, $\mu(E)< ...
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3answers
48 views

Continuous function $f$ on $\mathbb R $ such that $f \notin L^1 (\mathbb R)$ but $f \in L^1([a,b]), a< b $

Give an example of a continuous function $f$ on $\mathbb R $ such that $f \notin L^1 (\mathbb R)$ but $f \in L^1([a,b]), a< b $ If $f \in L^1([a,b]), a< b$ that would mean that ...
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1answer
33 views

Proving weak convergence of random probability measures

I don't understand the following as I read along a proof in a paper: We denote by $\mathcal{P}({M})$ the space of probability measures on a metric space $M$, equipped with the weak topology. ...
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1answer
22 views

Show: $\int_{\Omega} f d\mu =\int_{]0,\infty[} \mu(E_t) d\lambda_1(t)$

I have troubles understanding one step in the solution for this task: Let $(\Omega,\mathcal{A},\mu)$ be a $\sigma$-finite measure and $f:\Omega \rightarrow [0,\infty]$ be a measurable function. Let ...
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1answer
38 views

A very simple example of the dominated convergence theorem

Let's consider functions defined on the interval $[1, \infty)$. Let $f_n(x) = 1/n^2$ for $1 \le x \le n^2 $, and $f_n(x) = 0$ for $x > n^2$. Clearly $f_n$ converges to the constant function $0$ ...
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1answer
16 views

If $u\in L^p(\Omega;\mathbb R^d)$, then $u_i\in L^p(\Omega)$ for all $i\in\left\{1,\ldots,d\right\}$. Does the reverse hold true?

Let $\Omega\subseteq\mathbb R^d$ and $u:\Omega\to\mathbb R^d$ be Borel measurable. Since $$|u_i|\le\left\|u\right\|_2\;\;\;\text{for all }i\in\left\{1,\ldots,d\right\}$$ we obtain ...
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1answer
33 views

Which assumptions on $Ω\subseteq\mathbb R^d$ do we need in order to show density of $C_c^∞(Ω)$ in $(L^p(Ω),\left\|\;\cdot\;\right\|_{L^p(Ω)})$?

Let $\Omega\subseteq\mathbb R^d$, $u\in\mathcal L^1(\Omega)$ and $$u_\varepsilon(x):=\frac 1{\varepsilon^d}\int_\Omega\rho\left(\frac{x-y}\varepsilon\right)u(y)\;{\rm d}\lambda(y)\;\;\;\text{for ...