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.

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14 views

Find the total variation of a function that involves Lebesgue differentiable functions

Suppose $f: [a,b] \rightarrow \mathbb{R} \in BV[a,b]$. Let $f_a(x) = f(a)+ \int_a^x f' dm$ be the indefinite integral of $f$. Show that $V_a^b f_a = \int_a^b |f'| dm$. Attempt: Since $f \in BV[a,b]$, $...
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15 views

Differentiate the Total Variation (Lebesgue integral)

Let $f: [a,b] \rightarrow \mathbb{R} \in BV[a,b]$. Let $f_a(x) = f(a)+ \int_a^x f' dm$ be the indefinite integral of $f$. Let $f_s= f - f_a$. Show $v(x)= V_a^x f_s$ is singular i.e $v'=0$ a.e. Attempt:...
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31 views

Bounded variation implies Lebesgue integrable?

Let $f: [a,b] \rightarrow \mathbb{R}$ be a bounded variation. Let $f_a(x) = f(a)+ \int_a^x f' dm$ be the indefinite integral of $f$. Let $g= f - f_a$. Show $g$ is singular i.e $g'=0$ a.e. Attempt: ...
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25 views

changing measure from Lebesgue measure to Radon measure

Let $f$ be a bounded measurable function defined on $[0,1]$. Let consider $f$ be a Radon measure Let $\phi \in C^{1}_{C}[0,1]$ I have this integral $$\int_0^{1}f(t)\phi^{'}dt$$ How can we change the ...
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1answer
23 views

Function that uniformly converges to 0, but doesn't converges to 0 in norm.

Is there any sequence of function ${f_n}$ that fulfill 3 condition? Each function ${f_n} \in L^2(0, \infty)$, for every $n$. ${f_n}$ converges uniformly to 0 that is: for every $\epsilon > 0$, ...
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10 views

Continuity of the integral of an $L^1$ function

Let $f\in L^1\left(\mathbb R\right)$, define $g$ by $$g\left(t\right)=\int_{\mathbb R}\left|f\left(x+t\right)-f\left(x\right)\right|\,\mathrm{d}x$$ Prove that $g$ is continuous at zero. I tried to use ...
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31 views

If $g : S \rightarrow \mathbb{R}$ is a continuous function which $\forall s\in S:g(s)\ge 0$ and $\exists s\in S:g(s)>0$, does $\int_{S}g>0$?

I'm trying to prove that if a continuous function $g : S \rightarrow \mathbb{R}$ verifies that $\forall s \in S : g(s) \ge 0$ and $\exists s \in S :g(s) > 0$ where S is an open set that has volume $...
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38 views

Is every continuously differentiable function also absolutely continuous?

Context: I have been struggling a little bit understanding the concepts of absolute continuity and its relation to the Fundamental Theorem of Calculus. The main issue is that I find the definition of ...
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13 views

Meaning of P(dw) [duplicate]

I'm reading Shreve's Stochastic Calculus for Finance. On page 35, he gives this example: Example 1.6.4. Recall Example 1.2.4 in which $\Omega = [0, 1]$, $\mathbb{P}$ is the uniform (i.e., Lebesgue) ...
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18 views

If $g$ is bounded and measurable and $\mu$ is obtained by taking the lim inf of $g$ approaching each point, is $\mu$ Riemann integrable?

Let $$g:[a,b] \to \mathbb{R}$$ be bounded and measurable. Define $$\mu = \lim_{\delta \to 0} \mu_\delta(x),$$ where $$\mu_\delta(x) = \inf \{g(y):y \in b_\delta(x) \cap [a,b]\}.$$ Is $\mu$ always ...
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36 views

Is the lower bound and the upper bound of a bounded and measurable function Riemann integrable?

If $f: [a, b] \to \mathbb{R}$ is bounded and measurable. Let \begin{align*} & m_{\delta}(x) \\ = & \inf \left\{ f(x) : x \in (x - \delta, x + \delta) \cap [a, b] \right\} \...
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48 views

Is there a function $f\in L_{1}[0,1]$ but $ \frac{d}{dx}\Big(\int_{0}^{x}\frac{f(s)}{(x-s)^{\alpha}}ds\Big)\not\in L_{1}[0,1]? $

Let $\alpha\in (0,1).$ For a given $f\in L_{1}[0,1]$ consider $$ \phi(x)=\int_{0}^{x}\frac{f(s)}{(x-s)^{\alpha}}ds, \,\,\,\,\,x\in [0,1]. $$ It is clear that if $\phi\in AC[0,1]$ then $\phi$ is ...
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33 views

Calculating $\int_{\mathbb{R}^n}e^{-i(x|y)}e^{-a(|y_1|+ \cdots + |y_n|)}dy$

I need to calculate $$\int_{\mathbb{R}^n}e^{-i(x|y)}e^{-a(|y_1|+ \cdots + |y_n|)}dy$$ $a>0, \forall y \in \mathbb{R}^n$ and $(x|y)$ denotes the usual scalar product in $\mathbb{R}^n$ I have think ...
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54 views

How do I know if this integral is bounded?

The function $u(x, y)$ satisfies Laplace's equation $$ u_{x x}(x, y)+u_{y y}(x, y)=0 \quad(x \in \mathbb{R}, y>0) $$ and satisfies the boundary conditions $$ u(x, 0)=f(x) $$ where $f$ is integrable,...
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1answer
51 views

Why is $\int_{\Omega}X^+\,dP = \int _{0}^{\infty}P(X^+ > t)\,dt$?

$X$ be a random variable and $P$ is a probability measure on $\Omega$. Is it true that $$\displaystyle{\int_{\Omega}}X^+\,\mathrm{d}P = \displaystyle{\int _{0}^{\infty}}P(X^+ > t)\,\mathrm{d}t\,, $$...
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1answer
29 views

Integration with respect to finite variation processes

Hey can anyone explain me (or recommend a book) how we construct stochastic integral of (not necessarily continuous) process $H$ with respect to process with finite variation (also not continuous)?
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21 views

Integration By parts for Lebesgue integrals

Given bounded distribution functions $F$ and $G$, on $[a,b]$, such that $F$ and $G$ do not share a common point of discontinuity, following is true. $$\int_{(a,b]} FdG +\int_{(a,b]}GdF\\ = F(b)G(b)-F(...
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28 views

What is wrong with this heuristic proof of the Lebesgue Differentiation Theorem

The Lebesgue Differentiation Theorem is obvious for continuous functions (as the ball gets small, $f(x)$ approaches a constant). To prove it is true for all measurable functions (up to a set of ...
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1answer
31 views

Show for $1>a>0$ that $(e^{x}-a)^{-1}\sin(x)$ is integrable over $[0,\infty)$

Show for $1>a>0$ that $(e^{x}-a)^{-1}\sin(x)$ is integrable over $[0,\infty)$ Any help would be super helpful!
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1answer
42 views

The Lebesgue Integeral of $f(x) = \operatorname{int}(x^{2})$ over $[0,2]$ (integer part)

I just finished working on this Lebesgue integration problem and I am having doubts regarding my work. Note that I will use the notation $\int_{a}^{b}f(x)dx$ to denote the Lebesgue integral in this ...
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1answer
28 views

On the common compact support for a convergent sequence

Let $f\in L^1$. I guess it is true that there are $f_n$ s.t. $f_n\rightarrow f$ in $L^1$ and $\mathcal{F}(f_n)$ has compact support for each $n$. Can I conclude somehow that there is a compact set $...
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24 views

Prove using this definition that for two independent random variables $X,Y$ we have that $P_{X+Y}=P_X*P_Y$

Let $ \mu $ and $ \nu $ be measures on $ (\mathbb{R},\mathcal{B}(\mathbb{R})) $. The convolution of $ \mu $ and $ \nu $ is denoted $ \mu * \nu $ and is defined as $$ (\mu * \nu)(B) = \int_{\mathbb{R}^...
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20 views

Solution Verification for Lebesgue Integral of a Piecewise Function

I have just finished the following problem for my measure theory course and wanted some feedback on my work: Suppose that $f: [0,1] \to \mathbb{R}$ is defined by letting $f(x)=0$ on the Cantor set $\...
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1answer
18 views

Is $g(x)=\int_a^x (x-s)^{-\alpha}f(s) \, ds\in AC[a,b]$ if $f\in L_1[a,b]$ and $\alpha\in (0,1)$?

Let $f\in L_1[a,b]$ and $\alpha\in(0,1).$ Is it true that the function $$ g(x)=\int_a^x \frac{f(s)}{(x-s)^\alpha}\,ds $$ is absolutely continuous on $[a,b]$? First, I thought it is true and I tried ...
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22 views

Function in $L^2(\Omega)$ is in $H^1(\Omega)$

I want to show the equivalence between the two statements: ($\Omega$ is a bounded open subset of $\mathbb{R}^n$) $v\in L^2(\Omega)$ is in $H^1(\Omega)$, There is a constant $C$ s.t. $$ |\int_\Omega v\...
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31 views

Sources for multiple Stieltjes integral

I'd like to study more on multiple Stieltjes integral and want to know which sources (books or papers) provide a detailed discussion of multiple Riemann–Stieltjes integral or multiple Lebesgue-...
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18 views

Show $\frac{1}{\log(x)}$ is in $L^p ((0,0.5))$ for $p \geq 1$

Show $\frac{1}{\log(x)}$ is in $L^p ( (0,0.5))$ for $p \geq 1$ without using the $\operatorname{Li}(x)$ function or using $u = \log(x)$ substitution (that approach gives an infinite series and is not ...
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2answers
45 views

Integral comparison test for $\frac{1}{x \ln(x)^2} $ or $\frac{1}{x \log(x)^2}$ on $(0, \frac{1}{2})$

I am trying to show $f(x) = \frac{1}{x \ln(x)^2}$ on the interval $(0,\frac{1}{2})$ is in $L^p$ only for $p=1$ and not for $p>1$. I can show $||f||_{L^1( (0,\frac{1}{2}) )} < \infty $ but for $p&...
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1answer
36 views

Difference between a measurable set in $\mathbb{R}$ and $\mathbb{R}^n$ (Folland Theorem 2.40)

In the proof of Theorem 2.40 b), it says the proof follows from Theorem 1.19. But in Theorem 1.19, we have $E \subset \mathbb{R}$, not $\mathbb{R}^n$. So why does Theorem 2.40 still hold with $E \...
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17 views

Limits of an Integral: $\lim_{n\to \infty} \int x^2 \sin\left(f_n(x)\right)\mathrm{d}x=0$

I have a problem with the following exercise. I think by the Lebesgue dominated convergence theorem it can be solved but i don't know that $\sin\left(f_n(x)\right) x^2$ dominated by ? It it true to ...
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1answer
57 views

Is there a definition of Riemann integration in $\mathbb{R^n}$?

I know that Riemann integration is well-defined for a function $f: \mathbb{R} \to \mathbb{R}$. I would like to ask whether there is a definition of Riemann integration for a function $f: \mathbb{R^n} \...
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1answer
82 views

Does this integral converge when $\frac{1}{p}+\frac{1}{q}\geq1$

Fix a small $\delta>0$ and let $p,q>1$. Consider the integral $$I(p,q):=\int\limits_{1-\delta}^{1+\delta} \int\limits_{y/2}^{2y}\frac{1}{|y-x|^{\frac{1}{p}}|1-x|^{\frac{1}{q}}} \,\mathrm{d}x\,\...
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41 views

Exercise 5.S. the elements of integration and lebesgue measure

I'm having problems with this exercise. I've tried to apply the DOMINATED CONVERGENCE THEOREM but I couldn't. Could someone gives me any hint? Suppose the function $x\rightarrow f(x,t)$ is $X$-...
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17 views

Quantitative criteria of Lebesgue Integrability on $\mathbb{R}^d$.

Suppose that $f : \mathbb{R}^d \to \mathbb{C}$ is a Lebesgue measurable function on $\mathbb{R}^d$. Then a famous results are following : if $f$ satisfies \begin{equation} |f(x)| \lesssim \frac{1}{|x|^...
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2answers
87 views

Show $f: [0,1] \rightarrow [0, \infty)$ is Lebesgue measurable.

Let $X = [0,1]$. Let $f: X \rightarrow [0, \infty)$. Define $G(f) = \{ (x,y)\in X \times [0, \infty] : y \leq f(x) \}$. Show that $f$ is $m$-measurable iff $G(f)$ is $m^2$-measurable and that $$m^2(G(...
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34 views

Find a function $f$ such that $f \star f = g$ for a given $g$.

Let $g(x)=\frac{1}{x^2 |\ln x|}$ for $x \in ]0,a]$ with $a$ fixed in $]0,1[$. We know, thanks to Bertrand's integrale convergence, that $g \in L^1([0,a])$. I'm looking for a positive measurable ...
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2answers
117 views

The reason for the superiority of Lebesgue measure / Lebesgue integration

I'm learning about Lebesgue measure and Lebesgue integration. I know that Lebesgue integration is superior to Riemann integration in the sense that we can integrate a much larger class of function ...
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1answer
23 views

Fatou’s lemma equality

I found a sequence of functions that makes Fatou’s lemma be a strict unequality, but I’m not able to find one that makes it be an equality. Can you help me to find one? Thank you
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1answer
46 views

How to prove that $\int_{[-\pi,\pi]}\log(\vert 1- \exp(it)\vert)\mathrm{d}\lambda(t)=0$?

Let $r>1$ and $\int_{[-\pi,\pi]}\log(\vert 1- r\exp(it)\vert)\mathrm{d}\lambda(t)= 2\pi\log(r)$. We want to prove that : $\lim \limits_{r\to 1} \int_{[-\pi,\pi]}\log(\vert 1- r\exp(it)\vert)\mathrm{...
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1answer
19 views

BV[a,b]$\cap$C[a,b]$\neq$AC[a,b]

Take an $f\in$BV[0,1]$\cap$C[0,1] e.g. the Cantor function. I take the Lebesgue Stiltigies measure of $f$: $$ \mu_f((a,b])=f(b)-f(a). $$ Now I have a finite positive measure, so I can do the Radon-...
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1answer
25 views

If $f$ is integrable, how to show that $\sum_{n=1}^\infty 2^n\lambda(\{\lvert f\rvert\ge 2^n\})<\infty$ [duplicate]

I want to show the following converse of an (apparently) common integration theory exercise. Suppose $f:[0,1]\rightarrow\mathbb R$ is integrable, then $$\sum_{n=1}^\infty 2^n\lambda(\{\lvert f\rvert\...
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1answer
27 views

How to show that $\int \lambda((A-x)\cap B) \, d\lambda = \lambda(A)\lambda(B)$?

Given measurable sets $A, B\subset\mathbb R$, how can we show that $$\int_\mathbb R \lambda((A-x)\cap B) \, d\lambda = \lambda(A)\lambda(B)$$ holds? I don't see how the integrand might be simplified. ...
2
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1answer
46 views

$L^p$ convergence implies $\int_{\mathbb{R}^d \setminus E} |f_n|^p < \epsilon$ for all $n$.

Question. Suppose $f_n, f \in L^p(\mathbb{R}^d)$ such that $f_n \to f$ in $L^p$-norm. Is it necessarily true that for all $n \in \mathbb{N}$, and any $\epsilon > 0$, there exists a $E \subset \...
3
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0answers
44 views

Alternative proof to Folland 6.19 (Minkowski's Inequality for Integrals)?

Theorem 6.19. In short, we want to show $$\left\|\int |f(\cdot, y)|\,d\nu(y)\right\|_p \leq \int \|f(\cdot, y)\|_p \, d\nu(y)$$ for m'ble $f$. Could we prove this using just the duality of the norm? ...
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50 views

Is it okay to change variables here? What is the proper justification?

Suppose that the integral $$\int_{\mathbb{R}} \left|\int_{\mathbb{R}} f(\phi(x),y) dy\right|^p dx$$ is finite and that $\phi$ is montone and differentiable almost everywhere. Can one change variables $...
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1answer
35 views

$f$ is integrable iff this weird simple function is integrable.

I want to show that if $f:X\rightarrow\mathbb R$ is measurable and $a>1$, then $f$ is integrable iff $$\sum_{n\in\mathbb {Z}}a^n\mu(\{a^n\le\lvert f\rvert < a^{n+1}\})<\infty.$$ If we define $...
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1answer
44 views

Let $A ⊆ E ⊆ R$, where $E$ is measurable and $A$ is non-measurable. Prove that $m^∗ (E − A) > 0$

I have been asked to prove a statement- Let $A ⊆ E ⊆ R$, where $E$ is measurable and $A$ is non-measurable. Prove that $m^∗ (E − A) > 0$. I used the Caratheordory criterion- For a measurable set $E$...
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1answer
18 views

Dropping an assumption on bounded convergence theorem

If $f_n$'s are uniformly bounded complex valued functions converging pointwise to $f$ and $E$ is of finite measure, then we know that $f$ is integrable on $E$ and $\int_Ef_n\rightarrow \int_Ef$. How ...
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1answer
25 views

When is $\int_{[0, \infty)}f.d\lambda_1 = \lim_{n\rightarrow \infty} \int_{[0, n]} f.d\lambda_1$

In my Probability course there are some lebesgue Integrals wrt. Lebesgue Measure. In the notes there is a theorem that says: Let $f : [a, b] \rightarrow \mathbb{R}$ be bounded. If $f$ is Riemann ...
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1answer
36 views

Kind of equality on the sphere in Holder inequality

Lets suppose p > 1 and q as $\frac{1}{p} + \frac{1}{q} = 1$ $g$ is function of $L^q(\Omega, A, \mu) $ We suppose that exists $M$ positive number as : $M = sup\left \{ \int fg d\mu ; f \in \...

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