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.

learn more… | top users | synonyms

0
votes
0answers
39 views

Find the $\lim\limits_{c \to \infty} \int_{-\infty}^{+\infty} |g(x) - g(x+c)|dx$

Find the $$\lim_{c \to \infty} \int_{-\infty}^{\infty} |g(x) - g(x+c)|dx$$ where $g$ is integrable. I know already that if $g$ is integrable, than the integral of $g(x)$ and the integral of $g(x+c)$...
1
vote
1answer
65 views

Lebesgue and Riemann integrals two proofs

1. Let X be a finite closed interval [a,b] in R, let X be the collection of Borel sets in X and let λ be a Lebesgue measure on X. If f is a nonnegative function on X, show that ∫ fdu =∫a->b f(x)dx I ...
1
vote
1answer
38 views

how to prove isometric vector space isomorphism

Let $(L^1)^*$ be the dual space to $L^1$, or bounded linear functional over $L^1$, i.e., $f:L^1\to \mathbb{R}$, $f(cx+y)=cf(x)+f(y)$, $|f(x)| \leq M|x|^1$ Define the norm for $f \in (L^1)^∗$ by $||f||...
2
votes
0answers
35 views

Dominated convergence theorem vs continuity

Let $\{f_n\}$ be a sequence of functions in $L^2(0,1)$ such that $\lim_n f_n = f$ pointwise and $\vert f_n(x) \vert \leq g(x)$ for some integrable function $g$. By the dominated convergence theorem it ...
7
votes
2answers
86 views

Let $\int_{- \infty}^{\infty} f(x) dx =1$. Then show that $ \int_{- \infty}^{\infty} \frac{1}{1+ f(x)} dx = \infty.$

Let $f : \mathbb{R} \to [ 0, \infty)$ be a measurable function. If $\int_{- \infty}^{\infty} f(x) dx =1$. Then I want show that $ \int_{- \infty}^{\infty} \frac{1}{1+ f(x)} dx = \infty.$ Any help ...
1
vote
1answer
44 views

Show that $\mathcal{B}(X \times Y) \subseteq \mathcal{B}(X) \times \mathcal{B}(Y)$

Let $X$ and $Y$ be two locally compact Hausdorff spaces i. Show that each $f \in C_c(X \times Y)$ is a limit of sums of the form $$\sum\limits_{i=1}^n \varphi_i(x) \psi_i(y)$$ where $\varphi_i \in ...
2
votes
1answer
54 views

Exercise #9 in chapter 11 of Rudin's Principles of Mathematical Analysis.

Suppose $f$ is Lebesgue integrable on $[a,b]$. Let $F(x)$=$\int_{a}^x fdt$. Then prove that $F$ is continuous on $[a,b]$. I know that $F$ is continuous almost everywhere, because $F'(x)=f(x)$ ...
0
votes
0answers
18 views

Monotonicity property almost everywhere

I'm studying for a qualifying exam and having difficulty showing the following: Let $f\in L^1_{loc}(\mathbb{R})$ be a real-valued locally integrable function. Suppose that for each positive integer, $...
0
votes
1answer
25 views

Convergence of a series of functions almost everywhere

I'm studying for a qualifying exam and having difficulty showing the following: Let $f\in L^1(\mathbb{R})$. Show that $\sum_{n=1}^\infty n^{-1/2} f(x-\sqrt{n})$ converges absolutely for almost every $...
4
votes
2answers
45 views

Is $f(x)\exp(-x^2)$ summable if $f$ is square summable?

Suppose that $f \in L^2(\mathbb R)$; i.e. $$ \int_{- \infty}^\infty \vert f(x) \vert^2 dx < \infty. $$ Can we from this infer that $$ \int_{- \infty}^\infty \vert f(x)\vert e^{-x^2} \, dx < \...
0
votes
1answer
43 views

Differentiable almost everywhere of antiderivative function

Give an integrable function $h$ on $[a,\,b]$. Let $f(x) = \int_{a}^{x} h$ for all $x\in [a,\,b]$. Prove that $f$ is differentiable almost everywhere on $(a,\,b)$\ My attempt: I tried to show that $f$ ...
1
vote
1answer
84 views

Completing the proof that $\int f \, d\mu= \lim \int f_n \, d\mu$

The problem and solution is presented as above, but it doesn't seem that the proof is complete. Could someone show in details why the last step implies that $\int f \, d\mu = \lim \int f_n \, d\mu$ ...
2
votes
2answers
84 views

Lebesgue integrability implies finite almost everywhere

Let $(X,\mathcal{M},\mu)$ be any positive measure space. Suppose $f \in \mathcal{L}(X,\mathcal{M},\mu)$. Prove that $f(x)$ must be finite $\mu$-almost everywhere. I have defined a set $$E_n= [{x\in X ...
0
votes
1answer
24 views

Showing that a function converges in measure

Let $m(E) < \infty$. For the measurable functions $g$ and $h$ on $E$, define $\rho(g,h) = \int_E \frac{|g-h|}{1+|g-h|}$. , where $\int$ denotes the lebesgue integral. Show that $\{f_n\} \...
1
vote
1answer
63 views

Show that the conclusion may fail if the condition: $\lim \int f_n \, d\mu\lt +\infty$ is dropped.

Suppose that $(f_n)\in M^+(X, X)$ and $(f_n)$ converges to $f$, and that $$\int f \, d\mu=\lim \int f_n \, d\mu\lt +\infty,$$ we have the conclusion that $$\int_E f \, d\mu=\lim \int_E f_n \, d\mu$$ ...
0
votes
1answer
41 views

The Lebesgue Fundamental Theorem of Calculus

Let $f,g:\mathbb{R}\to\mathbb{R}$ be two Lebesgue integrable functions. If we have $$f(b)-f(a)=\int_a^bg(x)dx$$ for almost all $a,b\in \mathbb{R}$. How can we modify $f$ on a set of measure zero to ...
1
vote
1answer
47 views

Show there is a continuous isomorphism $l_{\infty}\rightarrow \left(l_1\right)^* $

Let $\left(l_1\right)^*$ be the dual space to $l_1$. Each $f \in \left(l_1\right)^*$ is a continuous linear functional over $l_1$. There is constant $C \in \Bbb R$ such that $|f(x)|\le C|x|_1, \forall ...
2
votes
2answers
138 views

What is $\displaystyle\int_{2}^{2}\frac{dx}{x-2}$?

Evaluate the integral: $$\displaystyle\int_{2}^{2}\frac{dx}{x-2}.$$ 1)When does $\displaystyle\int_a^a f(x)dx=0$? Always? 2)Does $\displaystyle\int_a^a$ means area between $(a,a)=\emptyset$? 3) Do ...
1
vote
2answers
46 views

Show $f$ is Lebesgue-Integrable

Let $f$ be a function defined over $[0,1]$, $$f(x) = \begin{cases}1 , & x\in \Bbb R-\Bbb Q \\ 0, & x\in \Bbb Q\end{cases}$$ Show that $f$ is Lebesgue-Integrable but not Riemann-Integrable. I ...
1
vote
2answers
40 views

Show that $\{m\in X:f(m)\ge g(m) \}$ is measurable

Show that if $f$ and $g$ are two (extended) measurable functions over ($X,\sigma(X),\mu$), then $\{m\in X:f(m)\ge g(m) \}$ is measurable. I know that in order to have $f:X\rightarrow \Bbb R \bigcup \{...
0
votes
1answer
41 views

Show that $L^1(\Bbb N)$ is the space of sequences whose series is absolutely convergent

Let $\mu$ be the counting measure on $\Bbb N$ whose sigma-algebra is its power set. Show that $L^1(\Bbb N)$ is the space of all real sequences ($a_n$) whose infinite sums are absolutely convergent. I ...
1
vote
1answer
17 views

For what values of $0 < p,q < \infty$ is the following inequality of integrals valid?

Let $m$ be the Lebesgue measure over $\mathbb{R}$ and let $f$ and $g$ be two nonnegative measurable functions defined on $[0,1]$ such that $f(x)g(x)\geq 1 \quad \forall x \in [0,1]$. It is not ...
0
votes
1answer
27 views

Does the Monotone Convergence Theorem apply?

I have a doubt about this exercise, that asks if the MCT apply. I think it does, 'cause the hypothesis holds. But I'm not completly sure about it. Can you give me a hint about how to prove it? ...
1
vote
2answers
17 views

Lebesgue integral - definition of the domain of the simple function.

The Lebesgue integral is defined as, $$\int f \, d\mu = \text{sup}\, \Big\{ \sum_{z\in s(M)} z\,\mu \,\big(\text{pr_im}_s(\{z\})\big) \Big\}$$ or the supremum of the sum of the areas under the curve ...
1
vote
0answers
36 views

Integral of a function defined on dense subset

[Edited] For a real-valued continuous function $f$ defined on a Lebesgue measurable dense subset of $[0,2]$, consider an integral $$ \int_{[0,1]}\frac{f(s)}{\sqrt{1-s}}ds. $$ My question is whether ...
3
votes
1answer
62 views

Lebesgue Integrable - graphical concept

I am having problems visualizing the process that determines whether a function is Lebesgue integrable. From A Garden of Integrals by Frank E. Burk: If a function $f$ is bounded measurable on the ...
2
votes
0answers
19 views

Moment problem for order statistics

Problem: Let $X_{1},\cdots,X_{n}$ be samples from $X$. If $E\left|X\right|^{a}<\infty$ for some $a>0,$ and $n,k,r$ satisfies $r\leq a\cdot\min\left(k,n-k+1\right),$ then $E\left|X_{\left(k\...
0
votes
1answer
40 views

Is $1/x$ integrable over the Cantor set? What about other not Lebesgue integrable functions and sets with Lebesgue measure $0$

I don't know how to integrate functions over fractal sets, so I would appreciate some pointers or references on the topic. What I want to know - are there cases when a function which is not ...
1
vote
2answers
26 views

change of variables in Lebesgue Stieljies integration

Suppose a R.V has distribution function $F$. Then I believe I can write $$\int_{-\infty}^0|x|^tdF(x)=\int_{0}^{+\infty}y^tdF(-y).$$But seriously, I don't know how to rigorously justify such formula. ...
2
votes
0answers
108 views

Calculation of a double integral [closed]

Can you help me to calculate this integral. $$\begin{align}I&=\int_{0}^{1}\frac{1}{x^2+1}\left(\int_{0}^{1}\frac{dy}{1+xy}\right)dx\\ &=\int_{0}^{1}\frac{ln(x+1)}{x(x^2+1)}dx\\ &=\ldots\...
0
votes
0answers
27 views

$0 \leq f \leq g$ measurable functions $\implies \int_g \ d\mu = \sup\limits_{f \leq s \leq g} \int s \ d \mu$.

Let all Lebesgue integrals be taken over the whole space $X$. Given two measurable functions $f,g : X \to [0, \infty]$ then it is enough to compute the Lebesgue integral supremum with simple ...
3
votes
1answer
24 views

Absolute continuity on limit

In probability space $(\Omega,\mathcal{F},P)$, we have random variable $X$ that maps $\Omega$ into $R$. Assume $\mathbb{E}{|X|}<\infty$ and let $A_n$ be a sequence in $\mathcal{F}$ satysfying $\...
2
votes
1answer
27 views

If $f^p\in L^1([0,1])$ it's bounded a.e.

We know that being Lebesgue integrable does not imply boundedness of the function (e.g. $g(x)=\frac{1}{\sqrt x}$). However function in $L^p$ spaces are functions with some decay conditions. Suppose ...
0
votes
1answer
30 views

Notation question: integral of sets? What does this mean?

I've been looking at this for a proof of Rademacher's theorem on differentiability of Lipschitz functions. At the start of p. 8, in the proof of the theorem, the pdf sets: $$A=\int_kA_k.$$ But that'...
3
votes
2answers
61 views

Let $f : E \rightarrow \mathbb{R}$. Show that if $|f|$ is measurable on $E$ and the set $\{f > 0\}$ is measurable, then $f$ is measurable on $E$.

I'm learning about Measure Theory (specifically measurable functions) and need help with the following problem: Let $f : E \rightarrow \mathbb{R}$. Show that if $|f|$ is measurable on $E$ and the ...
2
votes
2answers
104 views

Deduce that $f=0$ a.e.

Let $f:\Bbb R\to \Bbb R$ be a bounded Lebesgue measurable function such that for all $a,b\in \Bbb R$ with $-\infty<a<b<\infty$ we have $\int _a^b f=0$. Deduce that $f=0$ a.e. If $U$ is ...
1
vote
1answer
26 views

Showing $u'=v$ a.e. given $u_k \to u$ and $u'_k \to v$ in $L_2(\mathbb{R})$.

Suppose $(u_k)$ is a sequence of differentiable functions in $L_2(\mathbb{R})$ satisfying (1) There is a $u \in L_2(\mathbb{R})$ so that $\| u_k - u\|_2 \to 0$. (2) There is a $v \in L_2(\mathbb{R})$...
0
votes
2answers
25 views

From the whole space to sunsets: how to prove this integral limit?

Given $(X,\mu)$ a measure space, $\{f_n\}$ is a sequence of measurable and non-negative functions defined on $X$ such that $f_n\to f$ pointwise and $$\int_X f\mathrm d \mu=\lim_{n\to\infty}\int_X f_n\...
0
votes
1answer
48 views

Proving $f$ is constant almost everywhere when $f$ is bounded and measurable on a positive-finite measurable subset $X\subset\mathbb{R}$

I've been looking everywhere for help (and I've haven't got very far as of yet [!]), and I unfortunately don't have much insight in terms of starting this proof. The problem-statement is given as ...
0
votes
1answer
51 views

Calderón-Zygmund theorem doesn't seem has correct hypotesis

This theorem states that, given ANY $f\in L^1(\Bbb R^n)$ and ANY $\alpha>0$, there exists a sequence of (mutual disjoint open with sides parallel to the axis) cubes $\{Q_k\}_{k\ge1}$ such that $$ \...
0
votes
1answer
29 views

What's the adjoint for the evaluation operator?

What's the adjoint for the evaluation operator, $A\in L([X\rightarrow Y],Y)$, where $Af=f(x)$ for some fixed $x\in X$? In case, there's any ambiguity, I'm looking for an operator $A^*$ such that $\...
0
votes
1answer
26 views

Let $a_n$ be a sequence of nonnegative real numbers. Let $E = [1,\infty)$, and $f =a_n$ if $n\leq x<n+1$.Show that $\int_Ef = \sum a_n$

I have the following problem; Let $\{a_n\}$ be a sequence of nonnegative real numbers. Define the function $f$ on $E = [1,\infty)$ by setting $f(x) = a_n$ if $n\leq x<n+1$. Show that $\int_Ef = ...
0
votes
2answers
80 views

If $f$ is a bounded function on an interval E, and E has measure $0$ , Is $f$ measurable? What is the value of it's $\int_E f$?

If $f$ is a bounded function on an interval E, and E has measure $0$, Is $f$ measurable? What is the value of $\int_E f$? I have the question above in Royden Analysis 4e. Intuition suggests that f ...
0
votes
1answer
48 views

Is $\int f$ on [0,1] always equal to $\int f$ on [x,1] when take the limit of x to 0

Is $\int f$ on [0,1] always equal to $\int f$ on [x,1] when take the limit of x to 0? I know that if f is nonnegative, then I can use LMCT to prove it. However, how about f is only bounded, or only ...
2
votes
1answer
40 views

Why does $f(0)=\int f(x) \, d\mu =\frac14$ and $f$ extremal at $0$ imply $\mu(\{0\})=1$?

I just read a paper by Abner Shimony (The Status of the Principle of Maximum Entropy), and he is making a claim about a Lebesgue integral in appendix A that I don't fully understand yet. Let $f(x)$ ...
2
votes
1answer
113 views

Show that the integral of a bounded measurable function of finite support is properly defined.

Edit: My post might be lacking context. The question was posed because the book has only defined the Lebesgue integral of functions on a measurable set $E$ where $m(E)<\infty.$ The definition below ...
0
votes
0answers
35 views

A decreasing sequence of integrable functions with the limit of the integral exist

Question: Prove or disprove: If {${f_{n}}$} is a decreasing sequence of integrable functions such that $lim\int f_{n}$ exists in $\Bbb R$, then $lim\int f_{n}=\int lim\ f_{n}$. Attempt: I think this ...
0
votes
3answers
58 views

What is the integral of the function $f = \chi_{[0,\infty)}e^{-x}$

I have $$\int_{\mathbb R} f\,d\mu = \lim_{n\to\infty}\int_{[0,n]} f\,d\mu$$ So $$\begin{align}\int_{\mathbb R}f\,d\mu &= \lim_{n\to\infty}\int_{[0,n]}\chi_{[0,\infty)}e^{-x}\\ &=\lim_{n\to\...
2
votes
1answer
27 views

Why does this function preserve measure of null sets?

In this question, one of the answers claimed that the function $f: \mathbb{R}^{2n} \to \mathbb{R}$ given by $f(x,y) = x-y$ pulls back Lebesgue null sets to null sets, that is, $f^{-1}(N)$ is a null ...
3
votes
1answer
47 views

Monotone Convergence Theorem and Completeness of measurable space

I am reading about the monotone convergence theorem which states: Let $(X,\Sigma,\mu)$ a complete measurable space. If $f_n\rightarrow f$ monotone increasing and for almost all $x\in D$ then $\...