Questions tagged [lebesgue-integral]

For questions about integration, where the theory is based on measures. It is almost always used together with the tag [measure-theory], and its aim is to specify questions about integrals, not only properties of the measure.

Filter by
Sorted by
Tagged with
0 votes
0 answers
42 views

Evaluating the limit of the Lebesgue integral $\int_0^{2n\pi}\frac{(n+x)\sin(x/n)}{x(1+x)^2}\,dx$

I am stuck on this integral $$\lim_{n\to\infty}\int_0^{2n\pi}\frac{(n+x)\sin(x/n)}{x(1+x)^2}\,dx$$ I have learned MCT and DCT, but I don't how they might be applicable. Any hint would be appreciated.
user avatar
  • 123
-2 votes
0 answers
19 views

Does pointwise convergence imply measurability? and why?

$E$ is a Banach space and for every $u_0 \in E$ and $\{u_n\} \subset E$ with $u_n \to u_0$, we have $$S(t)u_n \to S(t)u_0$$ pointwise in $t \geq 0$. Why $t \mapsto S(t)u_0$ is measurable there ?
user avatar
2 votes
0 answers
49 views

Question about applying Dominated Convergence Theorem

Question: $\phi_n(x)=\int_{x_0}^xf(t,\phi_n(t))dt$, where $\phi_n(x)$ is continuous on $[a,b]$ and $f$ is continuous and bounded on $[a,b]\times(-\infty,+\infty)$. If $\phi_n(x)$ converges uniformly ...
user avatar
  • 401
3 votes
0 answers
40 views

Clarifying weird integral notation

I would like your help to understand the following notation for integrals which I have never seen. Consider an integral $$ \int_{a}^b 3 \text{ }d X $$ where $X$ is a random variable. What does this ...
user avatar
  • 242
0 votes
1 answer
35 views

Derivation under the sign of the integral (Lebesgue integral) using existence of bounded partial derivative [closed]

Let $f(x, y)$ be measurable function of two variables: $a < x < b$, $c < y < d < \infty$, such that there exists partial derivative $\frac{\partial f}{\partial x}$ which is bounded on $...
user avatar
  • 51
1 vote
1 answer
26 views

If $f \in L^1_{loc}$ then $\lim_{r \rightarrow 0} A_r f(x) = f(x)$ for a.e. $x \in \mathbb{R}$.

This is Theorem 3.18 in Folland. I am having trouble following one detail of his proof. His proof is as follows: It suffices to show that for $N \in \mathbb{N}$, $A_rf(x) \rightarrow > f(x)$ for a....
user avatar
  • 1,202
0 votes
1 answer
40 views

Showing function is well defined (measure theory)

Firstly apologies for the related sub questions in one post, but it felt better to give the complete question in one post rather then three separate posts repeating all the notation. Given a measure ...
user avatar
3 votes
2 answers
47 views

If $f \in C^1(\mathbb{R}, \mathbb{R})$ and $|f(x)| + |x||f'(x)| \to 0$ as $|x| \to \infty$, do we have $f'(x) \in L^1(\mathbb{R})$?

Let $f \in C^1(\mathbb{R}, \mathbb{R})$ such that $$|f(x)| + |x||f'(x)| \to 0, \qquad \text{as $|x| \to \infty$}.$$ Is it necessary that $f'(x) \in L^1(\mathbb{R}; \mathbb{R})$ with respect to ...
user avatar
  • 4,458
0 votes
1 answer
37 views

Does $\left| \int_a^b f dx \right| \le C$ for all $a$ and $b$ imply $f \in L^1(\mathbb{R}; dx)$?

Let $f : \mathbb{R} \to \mathbb{R}$ be Borel measurable such that there exists a $C > 0$ so that for all $-\infty < a < b < \infty$, $$\left| \int_a^b f dx \right| \le C,$$ where $dx$ ...
user avatar
  • 4,458
-1 votes
0 answers
31 views

Natural density of sets using Lebesgue measure

Suppose that the sets $A$ and $B$ are specified subsets of positive integers up to $n$. (For instance, $A$ or $B$ could be the set of all even integers less than $n$). Assume also that $A$ and $B$ ...
user avatar
0 votes
0 answers
24 views

$\int_E \lim \limits_{n\to\infty} f_n d\mu=\lim\limits_{n\to\infty}\int_E f_nd\mu$ , if $f_1\in \mathscr L^1(\mu,E)?$

Let $(X,\mathcal S,\mu)$ be a measure space and $E\in \mathcal S$. For every $n\in \mathbb N$, $f_n:X\rightarrow \overline{\mathbb R}$ is measurable with $f_1\leq f_2\leq \dots$. Then $\int_E \lim \...
user avatar
  • 77
3 votes
2 answers
40 views

Why does the monotone convergence fail here

I am looking at the example $f_{n}(x) = n \chi_{(0, \frac{1}{n}]}$. This converges to $0$ pointwise and graphing it out we can see that its a series of rectangles of area 1 but with growing height. I ...
user avatar
0 votes
1 answer
43 views

If $f$ non-negative and bounded and $\int_{\mathbb{R}}f d \lambda < \infty \Rightarrow \int_{\mathbb{R}}f^{2} d \lambda < \infty$

I am trying to show if $f$ is a non-negative function that is bounded and $\int_{\mathbb{R}}f d \lambda < \infty \Rightarrow \int_{\mathbb{R}} f^{2} d \lambda < \infty$ Where d$\lambda$ denotes ...
user avatar
-2 votes
1 answer
27 views

$\int_\mathbb Z 3^{-|x|}d\zeta(x)$ [closed]

Let $\zeta$ be the counting measure on $(\mathbb Z,\mathscr P(\mathbb Z))$. Calculate $\int_\mathbb Z 3^{-|x|}d\zeta(x)$. How can I calculate this integral?
user avatar
  • 1
0 votes
0 answers
23 views

Intersection of meaning between Lebesque Measure and Natural Density

FYI: the articles linked and their content within are well-known in number theory, so number theory is a tag (correct if need be). $\textbf{Background}$: This article by Maier states that a given ...
user avatar
0 votes
0 answers
19 views

$\int_E fd\mu=\lim \limits_{n\to\infty}\int_E f_nd\mu$

Let $(X,\mathcal S, \mu)$ be a measure space, $E\in \mathcal S$, $\mu(E)<\infty$ and $f_n$ be a non negative sequence, measurable real-valued functions that converge uniformly to $f:X\rightarrow \...
user avatar
  • 1
2 votes
2 answers
43 views

$L^2\subset L^1$ in case if the measure is finite [closed]

Let $(X, \mathcal B, \mu)$ be a finite measure space. Prove that $L^2(X, \mathcal B, \mu) \subset L^1(X, \mathcal B, \mu)$. I will be glad for any idea, comment, hint or advice.
user avatar
  • 35
1 vote
1 answer
24 views

Evaluate integral wrt Lebesgue measure and find the L^p space if it exists

I have been given the following the question: Consider the following function $f : \mathbb R → \mathbb R$, $$f=2·1_{(-3,1]}-3·1_{[5,+\infty)}$$ Here $1_A$ denotes the indicator function of set A. ...
user avatar
2 votes
3 answers
97 views

Square of a function vanishing outside bounded interval implies function is integrable

Prove that if $f$ is a non-negative function that vanishes outside a bounded interval, then $\int_{\mathbb{R}}f^{2}d \lambda < \infty \Rightarrow \int_{\mathbb{R}}f d \lambda < \infty$ Attempt: ...
user avatar
0 votes
2 answers
46 views

Measure theory showing integral of non-negative function is continuous

Let $f:(\mathbb{R}, B(\mathbb{R})\rightarrow(\mathbb{R},B(\mathbb{R}))$ be a non-negative function and $\int_{\mathbb{R}}f d \lambda < \infty$. $F:\mathbb{R} \rightarrow \mathbb{R}, F(x):= \int_{(- ...
user avatar
1 vote
0 answers
21 views

Is this function continuous (when $U$ has no flat regions)?

Apologies for that useless modifier in the brackets in the title -I had to add that to avoid "duplicate titles". It is most natural that there will be multiple questions with the title "...
user avatar
  • 1,047
1 vote
0 answers
19 views

Identity not integrable on surface measure

In my textbook in the definition of center of mass there is a following assumtpion: Let $S \in \mathbb R^m$ be a differentiable manifold such that $l_S(S) < \infty $ and the identity map on $\...
user avatar
0 votes
1 answer
29 views

Is any Borel measurable function in the function space $\mathscr{L}^2$?

I want to prove the relation $$\tag{3} \left|\int_{\mathbb{R}} \tilde{u}\tilde{v}\ d \lambda\right| \leq\left(\int_{\mathbb{R}}|\tilde{u}|^{2}d \lambda\right)^{\frac{1}{2}} \cdot\left(\int_{\mathbb{R}}...
user avatar
  • 475
0 votes
0 answers
36 views

Convergence of multiplication of two integrals [closed]

How to determine the convergence of the following construction $$\left(\int_a^{\infty}f(x)dx\right) \left(\int_0^{a}g(x)dx\right)?$$ In other word: which conditions are on $f$ and $g$ should be such ...
user avatar
0 votes
0 answers
51 views

Properly express two non-decreasing functions do not increase simultaneously

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ and $g:\mathbb{R}\rightarrow \mathbb{R}$ be two non-decreasing right-continuous functions. I have seen in an article where it is written that for all $t\geq 0$ ...
user avatar
  • 148
0 votes
0 answers
21 views

number of functions in$\mathscr L^1(\mu)$

For $(\mathbb N,\mathscr P(\mathbb N))$ look at the measure $\mu: \mathscr P(\mathbb N )\rightarrow \overline{\mathbb N}$, $A\mapsto 0\text{ if } A=\emptyset$ and $\infty, \text{ if} A\neq \emptyset$ ...
user avatar
  • 1
2 votes
1 answer
47 views

Existence of integral, $\int_{[1,\infty)} fd\nu$, with respect to measure, $\nu$.

Consider the Borel measure, $\nu$, on $[1,\infty)$ given by $$\tag{1} \nu(A):=\int_A \frac{1}{x}d\lambda, $$ where $\lambda$ is the $1$-dimensional Lebesgue measure. Let $f:[1,\infty)\rightarrow \...
user avatar
0 votes
1 answer
28 views

Is the measure $\nu(A):=\int_A \frac{1}{x}d\lambda$ $\sigma$-finite.

Consider the Borel measure, $\nu$, on $[1,\infty)$ given by $$ \nu(A):=\int_A \frac{1}{x}d\lambda $$ where $\lambda$ is the 1-dimensional Lebesgue measure. I want to know if this measure is $\nu$-...
user avatar
-3 votes
0 answers
29 views

$\int_D fd\mu=\int_D f_{|D}d\mu_{|\mathscr S_{|D}}$.

Let $(X,\mathscr S, \mu)$ bei a measure space and $D\in \mathscr S$. Furthermore, let $f: X \rightarrow \overline{\mathbb R}$ be a non negative, measurable function. Look at the function $f_{|D}:(D,\...
user avatar
  • 1
3 votes
2 answers
154 views

Volumen of the intersection of a cone and a tangent cone.

the professor of our integration class gave us the following exercise as an assignment: Determine the measure in $\mathbb{R}^3$ of $A=\{(x,y,z) \in \mathbb{R}^3 ,x^2+y^2+z^2-2x+2z\leq0\leq x^2-y^2-z^2 ...
user avatar
  • 553
0 votes
0 answers
16 views

A interesting question about relation between integral of composition(log) of a function and composition(log) of integral of the the function??

Let f:X-->R is a nonnegative measurable function and logf is integrable over X.(where X is a measure space) then to show this following two problem..i am completely stuck in this two problem how to ...
user avatar
1 vote
0 answers
32 views

How to show that $ (1+|x|)^{-M}$ is lebesgue integrable for M>n?

Is it true that $(1+|x|)^{-M}$ is lebesgue integrable for M>n over $R^n$ and if so how can we prove it? It seems like the theoretical definition for Lebesgue integral which is the sup of all simple ...
user avatar
  • 97
0 votes
0 answers
31 views

$\int_E f d\mu=\int_E f_{|E}d\mu_{|\mathcal S_{|E}}$

Let $(X,\mathcal S,\mu)$ bei a measure space, $E\in \mathcal S$ and $f:X\rightarrow \overline{\mathbb R}$ measurable and not negative. Show that $\int_E f d\mu=\int_E f_{|E}d\mu_{|\mathcal S_{|E}}.$ I ...
user avatar
  • 77
1 vote
1 answer
63 views

The limit of a Lebesgue integral

I'm trying to prove the following exercise: Suppose that $f,f_n,g\in\mathcal{L}^2(\mathbb{R})$, $n=1,2,...$, $f_n$ converges to $f$ $\mu-$almost everywhere, and $$ \int_{\mathbb{R}}|f_n(x)|^2 dx\leq 1 ...
user avatar
  • 73
1 vote
0 answers
46 views

Composition $F\circ g$ of Sobolev function $g \in W^{k,2}(\mathbb R^n)$ with smooth map $F$ is Sobolev

$\newcommand{\R}{\mathbb R}$ Let $F: \R\to \mathbb R$ be a smooth function such that $F(0) = 0$. Now consider $g\in W^{k,2}(\R^n)$ (Sobolev space) with $k>n/2$. Is it true that $F\circ g\in W^{k,...
user avatar
2 votes
1 answer
60 views

Can an "improper Lebesgue integral" differ from the ordinary Lebesgue integral?

A comment by Aloizio Macedo in this post claims claims Every "improperly Riemann integrable function" is also "improperly Lebesgue integrable". That Lebesgue doesn't know how to ...
user avatar
  • 4,869
1 vote
1 answer
26 views

Equivalence of the two definitions of integral of a non-negative function.

Let $(X,\mathcal S,\mu)$ be a measure space and $f:X\to [0,\infty]$ be a function.Then the first definition of integral of $f$ is as follows: $\int_X fd\mu=\sup\limits_{0\leq s\leq f,s \text{ simple}}...
user avatar
6 votes
2 answers
100 views

dense subspace of $L^2$ that is disjoint to $L^p$

I wonder if it is possible to have a dense subspace $U \subseteq L^2$ that is disjoint to $L^p$ for some $p\neq 2$. I would expect that such $U$ exists, but I'm stuck finding an example.
user avatar
2 votes
1 answer
55 views

When $\sum_{k=1}^\infty |a_k|<\infty,$ does $\int_0^{2\pi}\left(\sum_{k=1}^\infty a_k e^{ikx } \right)^n dx=0$ hold?

Let $a=\{a_k\}_{k=1}^\infty \subset \mathbb C$ satisfy $\sum_{k=1}^\infty |a_k|<\infty.$ Then, does $$\int_0^{2\pi}\left(\sum_{k=1}^\infty a_k e^{ikx } \right)^n dx=0\ \mathrm {for}\ n\in \mathbb N$...
user avatar
  • 2,221
6 votes
3 answers
86 views

Compute the following integral: $\lim_{n\rightarrow\infty}\int_0^{\infty}(1+\frac{x}{n})^{-n}\sin(\frac{x}{n})dx$

Question: Compute: $\lim_{n\rightarrow\infty}\int_0^{\infty}(1+\frac{x}{n})^{-n}\sin(\frac{x}{n})dx$. This is from Folland's Real Analysis book. If we can find an integrable majorant, then the ...
user avatar
  • 2,348
0 votes
1 answer
12 views

How to prove that the range of this integral operator $T$ is $L^2[a,b]$?

The operator is $Tf= g(t)+ \int_{a}^{t}(K(t,s)f(s)ds.$ The proof assumes that $g(t) \in L^2[a,b]$ so I believe I need to only prove that $\int_{a}^{t}(K(t,s)f(s)ds \in L^2[a,b]$. I started off this ...
user avatar
  • 128
0 votes
2 answers
56 views

Why does $\lim\limits_{n \rightarrow \infty} n \int^{x+1/n}_x F(t) dt = F(x)$ if F is continuous?

Why does $\lim\limits_{n \rightarrow \infty} n \int^{x+1/n}_x F(t) dt = F(x)$, where $F(x) = \int^x_a f(t) dt$ for a Lebesgue integrable function $f$. How do I use continuity of $F$ here? I know that $...
user avatar
1 vote
1 answer
39 views

Composition of measurable functions and measure

I was reading Lemma 1.22 in Kallenberg's Foundations of Modern Probability. It reads as follows: Lemma 1.22 (substitution). Fix a measure space $(\Omega, \mathcal{A}, \mu)$, a measurable space $(S, \...
user avatar
  • 11
1 vote
0 answers
27 views

Integral of Quantile Function is Expected Value?

Take a distribution function $F$. If $Q_F : [0, 1] \rightarrow \mathbb{R}$ is defined by $Q(p) = \inf\{x | p \leq F(x)\}$, is it true that $$E_F[X] = \int_0^1 Q_F d\mu$$ where $\mu$ is the Lebesgue ...
user avatar
3 votes
1 answer
94 views

Want to show that $f=0$ a.e.

This question has been asked before but I can't understand the given solution: To show that an integral is 0 a.e. if it is 0 over every subset of measure 2/3. Let $f\in L^1[0,1]$ such that for every $...
user avatar
  • 169
2 votes
1 answer
48 views

Lebesgue-Stieltjes integral and Dynkin $\pi-\lambda$ theorem

I am studying the Lebesgue-Stieltjes integral from this PDF: https://www.math.utah.edu/~li/L-S%20integral.pdf. In Theorem 8 the authors claim to use Dynkin's theorem in a way that I do not understand. ...
user avatar
  • 356
1 vote
0 answers
79 views

Find a relation between $a$,$b$ and $p$ so that $f(x)=\frac 1{(1+|x|^a)(1+\left|\log x\right|^b)}\in L^p\left(\mathbb R\right)$

A question in our measure theory test reads Let $f:\mathbb R\to \mathbb R$ such that $$f(x)=\frac 1{\left(1+|x|^a\right)\left(1+\left|\log x\right|^b\right)}$$ Find a relation between $a$,$b$ and $p$ ...
user avatar
  • 5,553
1 vote
1 answer
50 views

$f\in W^{1,2}(\mathbb R)$ and $\|xf(x)\|_2<\infty$ implies $\lim_{x\to\infty}x|f(x)|^2=0$

I'm trying to apply an integration by parts to solve the Exercise 8.18 at Folland's Real Analysis. But, for that, I need to have "$f\in W^{1,2}(\mathbb R)$ and $\|xf(x)\|_{L^2}<\infty$ implies ...
user avatar
  • 73
0 votes
0 answers
21 views

Integration and Measure problem from Shilov and Gurevich Book.(zero measure definition using step functions))

The problem reads: Let F be the closed interval obtained by removing a countable collection of disjoint open intervals from a closed interval [a,b], where the sum of the lengths of such intervals is b-...
user avatar
0 votes
1 answer
24 views

Prove that $\sup_x |\rho(x)-\rho_{\alpha}(x)| \leq |\rho'|_{L^{\infty}(\mathbb{R})} |1_{x>0}-h_{\alpha}|_{L^1}$

Let $\rho(x)$ be a function such that its derivative is bounded and its limit at $x \to -\infty$ exists. We can rewrite $\rho$ like so and define $\rho_{\alpha}$: $$ \rho(x)=\rho(-\infty)+\int_{-\...
user avatar

1
2 3 4 5
136