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

4
votes
2answers
50 views

Limit of integral over $[0,1]$ of $\frac{ne^{-x}}{1+nx}$

I need some help to calculate the following limit (in measure theory): $$\lim_{n \to \infty} \int_{0}^{1} \frac{ne^{-x}}{1+nx}dx$$ My first idea was to use either the monotone convergence theorem or ...
0
votes
1answer
27 views

In the theorem of coincidence of $R$ and $L$ integrals: are we assuming the $\Bbb L$ $\sigma$-algebra? If not, how to prove the measurability of $f$?

Theorem: If $f: [a,b] \to \Bbb R$ is Riemann-integrable, then $f$ is Lebesgue-integrable, and: $$ \int_a^b f(x) dx = \int_{[a,b]} fd\lambda$$ Here is the proof: (assuming that we know ...
1
vote
0answers
39 views

Will the joint probability density exist in this case?

Assume that we have a probability space $(\Omega, \mathcal{A}, P)$, and we have two random variables $X,Y: \Omega \rightarrow \mathbb{R}$. On this space. We can define two measures ...
0
votes
0answers
18 views

Upper bound for integral on some environment of zero

I'm trying to proof an estimate that should not be too hard to proof. Let $f$ be some integrable non-negative function and $c>0$ some arbitrary constant. I claim that there exists some ...
2
votes
1answer
66 views

Will the integral still be 1?

We can show that $\int_{\mathbb{R}}\frac{e^{-\frac{(x-m)^2}{2}}}{\sqrt{2\pi}}=1$. The easiest way is by observing that this is the probability density of a normal random variable, with mean m, and ...
1
vote
0answers
18 views

Criterion for a signed measure to be positive

I am reading a proof in which I do not understand the following claim: Let $K$ be a compact metric space and let $M(K)$ be the set of signed Borel measures on $K$. Then the set $P(K)$ of positive ...
1
vote
2answers
49 views

Chebyshev's Inequality in proof of Proposition 23 of Royden?

This is from the 4th edition of Royden, on page 92. Proposition 23. Let $f$ be a measurable function on $E$. If $f$ is integrable over $E$, then for each $\epsilon >0$, there is a $\delta ...
1
vote
2answers
68 views

Show $\int_\pi^\infty \frac{dx}{x^2 \left( \sin^2(x) \right)^{1/3}}$ is finite using 1st semester measure theory

I am studying for a real analysis qualifying exam, and I am completely stuck on this problem. Show $$ \int_\pi^\infty \frac{dx}{x^2 \left( \sin^2(x) \right)^{1/3}} $$ is finite. I tried the ...
1
vote
1answer
39 views

Measure Theory on integrals

Is it true that if $f\in L^1 (X,\mu)$ then $$\int_E |f| d\mu +\int_{E^c} |f| d\mu = \int_X |f| d\mu?$$
1
vote
1answer
64 views

On generical domains: Riemann integrable $\Rightarrow$ Lebesgue integrable?

If I have correctly understood and been able to generalise the proof that I found in Kolmogorov-Fomin's Элементы теории функций и функционального анализа for the case of $n=1$, I know that, if ...
0
votes
1answer
31 views

Example about Fubini/Tonelli Theorems

By using Fubini/Tonelli Theorems, evaluate $$\int^{0}_{1} \int^{1}_{y} x^{-\frac{3}{2}}\cos\left(\frac{\pi y}{2x}\right)dxdy$$ My attempt: by Tonelli Thm $$\int^{0}_{1} \int^{1}_{y} ...
3
votes
3answers
116 views

Calculate limit of integral sequence

Hi i need to calculate limit of integral sequence: ...
2
votes
0answers
30 views

Dense subspaces of the space $L^p(0,T;X)$

Given a Banach space $X$ and $1\leq p<\infty$, let's define the space $L^p(0,T;X)$ as the set of all strongly measurable functions $f:(0,T)\mapsto X$ such that $$\int_0^T\Vert ...
0
votes
2answers
27 views

If $f\in L^2$ then there is $g$ continuous with compact support s.t. $\int (f-g)^2<\varepsilon$

Let $f\in L^2$ how can I show that for all $\varepsilon>0$ there is $g$ continuous with compact support s.t. $\int (f-g)^2<\varepsilon$ ? I know that $f\in L^2\implies f^2\in L^1$ and since the ...
-1
votes
0answers
21 views

Boundedness of expectation

Suppose I have a sequence of random vectors $\{X_n\}_n$ defined on the probability space $(\Omega, \mathcal{F}, \mathbb{P})$, where $X_n:\Omega \rightarrow \mathbb{R}^h$. Each $X_n$ induces the ...
-1
votes
0answers
37 views

Maximal function [on hold]

If $f$ is integrable on $\mathbb{R}^d$, we define its maximal function $f^*$ by $$ f^*(x) = \sup_{x\in B} \frac{1}{m(B)} \int_B |f(y)| \; dy, \;\;\;x \in \mathbb{R}^d $$ where the $\sup$ ...
2
votes
0answers
52 views

$1 = \int f(x) \ dx$, by definition, or by Lebesgue's theorem?

We have that (in the context of Lebesgue integration)$$\lim_{n \rightarrow \infty} \int_{-\infty}^n f(x) \ dx = 1$$ I wish to show that this implies $\int_{-\infty}^\infty f(x) \ dx = 1$. Is this true ...
0
votes
1answer
65 views

Question about Lemma 5.9 in Royden “Real Analysis” 3rd, pages 106 - 107

The lemma will show that If $f$ is bounded and measurable on $[a,b]$ and $F(x) = \int^x_a f(t)dt + F(a)$, then $F'(x)= f(x)$ for almost all $x$ in $[a,b]$. I understand the basic idea of the proof, ...
-2
votes
1answer
63 views

Lebesgue measurable function question

Let $f$ be L measurable on [0,1] with $f(x)>0$ a.e. Suppose $A_{n}$ is a sequence of measurable sets in [0,1]. Prove that if $\lim_{n \to\infty} \int_{A_{n}}^{ } f(x) dx=0 $ then $\lim_{n\to\infty} ...
1
vote
0answers
48 views

Proving limit of sequence equals $0$

Let $f: \mathbb R \rightarrow \mathbb R$ be integrable and $a>0$. I would like to prove that $\lim_{n \rightarrow \infty} f(nx) / n^{-a} = 0$ for almost all $x \in \mathbb R$. We already ...
1
vote
0answers
53 views

convergence of series over $f(n+{x\over a})$ [duplicate]

let $f: \mathbb R\rightarrow \mathbb R$ be integrable and $a>0$. I have to show that for almost all $x \in \mathbb R$ the series $\sum_{n \in \mathbb Z} f(n+{x\over a})$ converges, but I have no ...
0
votes
0answers
22 views

$ \lim_{c\rightarrow a, d\rightarrow b} \int_{c}^{d} f(x)dx = \int_{a}^{b} f(x)dx $ if $f$ is integrable

I want to prove that if $f$ is integrable, then : $ \lim \limits _{c\rightarrow a, d\rightarrow b} \int_{c}^{d} f(x)dx = \int_{a}^{b} f(x)dx $ Because $f$ is integrable , we know that $ ...
0
votes
0answers
21 views

Deciding whether a function $f$, defined on $\mathbb{R}^2$, is Lebesgue-integrable over two sets

I want to decide on which of the following sets $A \subseteq \mathbb{R}^2$ the function $f(x, y) = e^{-x y} sin(x)$ is Lebesgue-integrable: $$(i) A = [0, R] \times [0, \infty], R > ...
0
votes
1answer
56 views

Explanation of a step in evaluating $\int_0^{+\infty} e^{-x^2} {\rm d}x$

The evaluation of the Gaussian integral is standard in Undergraduate Calculus ( Calculus II ?). I am not really asking how to evaluate this integral, which can be done by appealing to Fubini's ...
0
votes
2answers
59 views

Continuity of the function $f(y)=\int_{0}^{+\infty} y\sin(x) e^{-xy} \, dx $ in $y=0$

How do you prove the continuity of the follow function in $y=0$ $f:[0,+\infty) \rightarrow \mathbb{R}: f(y) =\int_0^{+\infty} y\sin(x) e^{-xy} \, dx$ We change the variables: $x\rightarrow \frac ...
3
votes
1answer
60 views

sufficient and necessary condition for an integral to be finite.

Suppose that $a_1,\ldots,a_n>0$ and consider the function $f:\mathbb R^n\rightarrow \mathbb R$ $$f(x_1,\ldots,x_n)=\frac{1}{x_1 ^{a_1}+\cdots+x_n ^{a_n}}$$ I'm trying to show that ...
2
votes
0answers
28 views

Expectation of a random vector and Fubini's theorem

I have a question related to the definition of expectation of a random vector, in particular, to its relation (if any) with Fubini's Theorem. Consider the random vector $X:=(X_1,X_2,X_3)$ of ...
0
votes
1answer
14 views

boundaries of Lebesgue-integral

Let $A=\{(x,y)\in\mathbb R^2|x\lt y\lt 2x, 1-x\lt y\lt 3-x\}$ be a set and evaluate $\int_A \frac{y}{x} d\lambda_2(x,y)$. I calculate that $x\in(\frac{1}{3},\frac{3}{2})$ and $y$ depends on x ...
0
votes
4answers
75 views

Calculate $ \lim _{n \rightarrow \infty }\int ^{n}_{-n} \left(\cos\left(\frac{x}{n}\right)\right)^{n^{2}}\,dx $

Calculate this limit: $$ \lim _{n \rightarrow \infty }\int ^{n}_{-n} \left(\cos\left(\frac{x}{n}\right)\right)^{n^{2}}\,dx .$$ We're given the follow inequality: $$r^{k} \leq \exp(-k(1-r)),$$ with ...
1
vote
1answer
67 views

Examples of theorems from advanced calculus that are easier to prove with Lebesuge integral.

I am looking for interesting examples of theorems in advanced calculus (at the level of Rudin's "Principles of Mathematical Analysis") that are easier to prove when using Lebesgue integration, or ...
2
votes
1answer
25 views

Determining two iterated integrals over the diagonal line of the cube $[0, 1]^2$

Consider the interval $[0, 1]$ with the Borel-$\sigma$-algebra $\mathcal{B}([0, 1])$. Let $\lambda$ be the Lebesgue-measure and $\mu$ the counting measure on $\mathcal{B}([0, 1])$. For the ...
1
vote
1answer
16 views

A positive measurable and improper integrable function is integrable

Let $f: \mathbb{R} \rightarrow [0,+\infty)$ a positive measurable function. Suppose $f$ is improper integrable. Show that $f$ is integrable. Take a row $x_{n} \rightarrow \infty$ and a row ...
0
votes
1answer
29 views

Exponential growth as check for integrability

I encountered this proposition: " $\int\limits_{-\infty}^{\infty} |f(t)| dt < \infty$, i.e. $f(t)$ has to grow slower than an exponential curve." Is exponential growth the slowest increment that ...
0
votes
1answer
8 views

Given $A\subset [0,1]$ prove that $m^*(\{1-x:x\in A \})=m^* (A)$

Given $A\subset [0,1]$ prove that $m^*(\{1-x:x\in A \})=m^* (A)$. ($m^*$ is outer Lebesgue measure) $$m^*(\{1-x:x\in A \})=\inf\{m^*(G):\{1-x:x\in A \}\subset G \mbox{ and } G \mbox{ is ...
2
votes
0answers
45 views

Showing Lebesgue's definition of measure implies measurability

19. Let $\mu^*$ be an outer measure on $X$ induced from a finite premeasure $\mu_0$. If $E \subset X$, define the inner measure of $E$ to be $\mu_*(E) = \mu_0(X) - \mu^*(E^c)$. Then $E$ is ...
0
votes
1answer
40 views

Example when $\int \sum f_k d\mu \not = \sum_k \int f_k d\mu$ and $\sum_k f_k $ absolutely convergent

Hi this is a problem I find and I have had problem finding the example which the problem ask I'd appreciate if someone can help me with this. Thank you. Let $(X,\mathscr A, \mu)$ a measure space ...
3
votes
3answers
54 views

Is the following function Lebesgue integrable?

Given is the function $$f(x)=\left\{ \begin{array}{rl} \frac{1}{\sqrt{x}}, & x\in \mathbb{I}\cap [0,1]\\ x^3, &x\in \mathbb{Q}\cap [0,1]\end{array}\right.$$ I have to see if $f$ is ...
2
votes
1answer
25 views

Existence of $L^{1}(\mathbb{R}^{n})$ Function Defined via Functional Equation

Perhaps, I'm reading the problem statement wrong, and it's not asking for existence, only uniqueness; but in any case... Problem. Let $g\in L^{1}(\mathbb{R}^{n})$, $\|g\|_{L^{1}}<1$. Prove that ...
3
votes
1answer
104 views

Showing Convergence of Positive Series

Problem. Let $\{A_{\vec{k}}\}$ be a sequence of real numbers indexed by vectors $\vec{k}=(k_{1},\ldots,k_{n})\in\mathbb{N}$. Let $\{r_{\vec{k}}\}$ be a sequence of positive real numbers such that ...
1
vote
0answers
37 views

Riesz-Fischer Theorem with Variable Exponent

Let $p:\mathbb{R}^{d}\rightarrow [2,4]$ be a measurable function (we can relax this to $[1,c]$, for any $c>1$), and let $f_{n}:\mathbb{R}^{d}\rightarrow\mathbb{R}$ be a sequence of measurable ...
3
votes
1answer
46 views

If $f:\mathbb{R}\to\mathbb{R}$ is Lebesgue integrable, then $\lim\limits_{t\to 0}\int_\mathbb{R}|f(x+t)-f(x)|dx=0.$

The problem is stated in the title. If $f$ is Riemann integrable, then the Dominated Convergence Theorem and Lebesgue's Criterion for Riemann integrability does it, however I am stuck when the ...
1
vote
2answers
22 views

Show that $f(x)=\int_{[a,x]} g \, dm$ is continuous on $[a,b]$

Let $g$ be summable on $[a,b]$. Show that $f(x)=\int_{[a,x]} g \, dm$ is continuous on $[a,b]$. ($m$ is Lebesgue measure on $\mathbb{R}$) First let $g$ be a simple function. For simplicity assume ...
2
votes
1answer
34 views

If $f_k\to f$ in measure, does $f_k\to f$?

I recall that 1) $f_k(x)\to f(x)$ if $\lim_{k\to \infty }f_k(x)=f(x)$ 2) $f_k\to f$ in measure if $$\forall \varepsilon>0: \lim_{n\to \infty }m(\{x\mid |f_k(x)-f(x)|\geq \varepsilon\})=0$$ 3) ...
4
votes
1answer
43 views

An integrable function is finite a.e.

Let $f:\mathbb R\longrightarrow \mathbb R$ an integrable function and let $F(x)=\int_{-\infty }^x f(t)dt$. 1) Show that $f$ is finite a.e. 2) Show that $F$ is uniformly continuous. What I tried: ...
0
votes
1answer
64 views

Prove a function vanishes almost everywhere.

This is an exercise from Natanson's Theory of Functions of a Real Variable. Suppose $f\in L^1[a,b]$, and $\alpha\in(0,b-a)$ is a constant. If for all $E\subset[a,b]$ such that $m(E)=\alpha$, we ...
0
votes
1answer
43 views

a Very simple example

When talking of functions that are Riemann integrable but not Lebesgue integrable we always give the example of $f(x)=\frac{\sin(x)}{x}$ on $]0,\infty [$ but my question is : is it the same with ...
1
vote
1answer
69 views

Let $f: [0,\infty) \to \mathbb R$ be a $\lambda$-integrable function such that $\int_{[0,t]} fd\lambda =0$ for all $t\ge 0$. Prove that $f=0$ a.e.

this is an exercise that I find. I'd like to know if the following is correct. Let $f: [0,\infty) \to \mathbb R$ be a $\lambda$-integrable function such that $\int_{[0,t]} fd\lambda =0$ for all ...
9
votes
0answers
173 views

$\int\limits_{\Omega}{uvdx}<\infty,\forall v\in H_0^1(\Omega)$ implies $u\in L^{6/5}(\Omega)$

Let $d=3$ and $\Omega\subset \mathbb R^d$ is a bounded Lipschitz domain and $u$ is a measurable function. A sufficient condition for the integral $\int\limits_{\Omega}{uvdx}<\infty,\forall v\in ...
0
votes
0answers
59 views

evaluating Lebesgue integral 3

Let $A_{R,r}$ be a set with $0\lt r\lt R$ $$A_{R,r}= \{(x,y,z)\in\mathbb R^3\mid (R-\sqrt{x^2+y^2})^2+z^2\le r^2\}$$ Evaluate $\lambda_3(A_{R,r})$ Answer: I used Cavalieri's principle and set ...
-3
votes
1answer
22 views

Use monotone convergence theorem to find $\int_{(0,1)} gdm$ where $g(x)=x^{-\frac{1}{2}}$. [closed]

Use monotone convergence theorem to find $\int_{(0,1)} gdm$ where $g(x)=x^{-\frac{1}{2}}$. Any idea which sequence of functions should I use? Edit: $m$ is standard Lebesgue measure on $\mathbb{R}$. ...