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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Riesz–Markov–Kakutani representation theorem on a $\sigma$-compact space

I'm trying to prove that given the conditions of the Riesz–Markov–Kakutani representation theorem and additionally assuming the space is $\sigma$-compact, the result measure is inner regular for every ...
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$g \in L^p_{\operatorname{loc}}(\Omega) \Longleftrightarrow |g|^p \, \chi_\Omega \in L^1_{\operatorname{loc}}(\mathbb R^n) ? $

Let $\Omega \subset \mathbb R^n$ be an arbitrary open set and consider the usual Lebesgue spaces of locally $p$-integrable functions, for $1 \leqslant p < \infty$. My goal is to prove the following ...
xyz's user avatar
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Let $I(x) := \int_{B(x,R)} \frac{f(y)}{|x-y|^{n-1}} dm(y)$. Then there exists $C = C(n)$ such that $|I(x)| \leq CR \cdot Mf(x)$

Let $\lambda_n$ be the $n$-dimensional Lebesgue measure, and let $f\in L^1_{loc}(\mathbb R^n)$ and $Mf$ be the maximal function of $f$. For fixed $R>0$ and $x \in \mathbb R^n$ and we define $$ I(x)...
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What is a good reference book for the topic "Approximation of Measurable Functions by Simple Functions" in measure theory? [closed]

I am having a course in Measure Theory this semester, where we're being taught Approximation of Measurable functions by Simple functions. Our professor generally refers to Stein-Shakarchi but can ...
Gourab Chowdhury's user avatar
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Questions about compact metric space $X$ with finite probability measures with finite supremum.

Let $X$ be compact metric space, and $\mu_n$ a sequence of finite Borel measures on $X$ such that $\sup_n \mu_n(X)<\infty$. Prove (a) For every $f \in C(X)$, there exists a subsequence $n_j$ such ...
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Convergence on $L^1$ space

I come across this problem while reading hints for Exercise 3.15 (Brezis): Let $\Omega=(0,1)$ and sequence $f_n$ defined by $f_n(x)=ne^{-nx}$. Prove that: $$\displaystyle \int_{\Omega} \varphi f_n\...
Hải Nguyễn Hoàng's user avatar
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Finding a Lebesgue integrable function for every $1 \leqslant q < \infty$ that satisfies aditional requirement.

Consider the usual Lebesgue spaces. Amid one of my studies, I started wondering if it is possible to find an example that satisfies the following problem: Problem. Consider arbitrary elements $1 \...
xyz's user avatar
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Prove that $\int f \ln(f) d \mu =\sup \left \{ \int f \phi d \mu : \int e^{\phi} d\mu \leq 1 \right \}$

Question Prove that $\int f \ln(f) d \mu = \sup \left \{ \int f \phi d \mu : \int e^{\phi} d\mu \leq 1 \right \}$ With $f$ verifying $ \int f d \mu = 1 $ and $ f \cdot \ln(f) $ is integrable, with $ \...
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Proving a given metric is a metric on the product of a measure space with itself.

Let $\mathbf{M}_1(X)$ denote the set of Borel probability measures on a compact metric space $X$. Given $\mu_1,\mu_2 \in \mathbf{M}_1(X)$, let $J(\mu_1,\mu_2)$ be the set of Borel probability measures ...
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Example of a function on $([01]; \sigma ([0;1]); \lambda )$ for which it has no meaning to write $ \int f d \lambda$

-I am trying to understand Lebesgues integration and in order to understand well this concept I would likte to have an example of a function $]0;1] $ ( or $ [0;1[ , [0;1] , ]0;1[ $ ) for which it has ...
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Subsequence which converges for compact metric space.

Let $X$ be a compact metric space and $\mu_n$ a sequence of finite Borel measures on $X$ with the property that $$\sup_n \mu_n(X)<\infty.$$ Show for all $f \in C(X)$ there exists a subsequence $n_j$...
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Lebesgue integral and limit; $\lim_{a\to 0^+}\int_{a}^{1}(t\ln(t))^3dt=\int_{0}^{1}(t\ln(t))^3dt$

How can I prove: $$\lim_{a\to 0^+}\int_{a}^{1}(t\ln(t))^3dt=\int_{0}^{1}(t\ln(t))^3dt$$ assuming that the integral are Lebesgue integrals. There may be a theorem that confirms this equality.
Minimus Heximus's user avatar
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Does there exist a subset $S$ in $[0,1]$ such that for all interval, $S$ and its complement in the interval has the same Lebesgue masure?

I would want to see a strengthening or a disproof of a result from an exercise from Rudin's Real and Complex Analysis, that is asked before at here Construct a Borel set on R such that it intersect ...
Emancipatrix's user avatar
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Define multiplication on $L^1(\mathbb{R})$

I want to define multiplication on $L^1(\mathbb{R})$, of course, convolution is an allowed multiplication which makes $(L^1(\mathbb{R}),*)$ is a Banach algebra, I want to know if there are other ...
InnocentFive's user avatar
2 votes
2 answers
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Approximating norm of a Hilbert space point with the norm of a vector

I have the Hilbert space of square integrable functions on $[a, b]$, and what I would like to have is to discretize this space, i.e., find a sequence of finite-dimensional Hilbert spaces $H_K$ of ...
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Question concerning the correctness of this version of Fatou's Lemma

In lecture we learned about Fatou's Lemma stated as follows: Let $(X, \mathcal{S}, \mu)$ be a measure space and $(f_k : X \to [0,\infty])$ measurable and $f: X \to [0, \infty] $ a function such that: $...
user007's user avatar
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Defining the expectation of a measurable function with respect to a (non-probability) measure

The typical definition of expectation requires a probability space and a random variable Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space, $(\mathsf{X}, \mathcal{X})$ be a measurable ...
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Laplace Transform of a Piece-wise function with a Weibull distribution.

Suppose I have the following piecewise function: $$Q(t) = \begin{cases} W(t) & t<T \\ 1 & t=T \\ 0 & t>T \end{cases}$$ ...
Keyvan's user avatar
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For $f \in H^s$, then $\exists g \in C_c$ such that $f=g$ a.e.

Consider the space $H^s(\mathbb R^d)$ ($f \in L^2$ not in Schwartz class), $s \in \mathbb R$. Apply Riemann-Lebesgue Lemma to $\hat{f}$ to show that for some $s>s_0$ then there is a continuous ...
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Confusion on Folland's redefinition of L^1 space.

The following are excerpts from Folland: Proposition 2.12: 2.12 Proposition. Let $(X, \mathcal{M}, \mu)$ be a measure space and let $(X, \bar{\mathcal{M}}, \bar{\mu})$ be its completion. If $f$ is an ...
juekai's user avatar
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1 answer
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Understanding the term almost everywhere in measure theory

I am trying to understand the term "almost everywhere" from measure theory correctly. So given two extended real-valued integrable functions $f, g: X \rightarrow \bar{\mathbb{R}}$ with $$\...
guest1's user avatar
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Sanity check - monotone increasing function on $[0,\infty)$ and integral of its derivative

Let $f : [0,\infty) \to \mathbb{R}$ be monotone-increasing. Then, it is well-known that the derivative $f'$ exists and takes nonnegative values almost everywhere on $[0,\infty)$. Then, is it true that ...
Keith's user avatar
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Is it possible to exploit higher integrability in the trace theorem?

Let $\Omega\subset\mathbb{R}^n$ be a bounded smooth domain and $n>1$ and let $T_p\colon W^{1,p}(\Omega)\to L^p(\partial\Omega)$ denote the trace theorem for $p\in[1,\infty]$. Suppose, we are given $...
Muschkopp's user avatar
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Bounds on volume of set with orthonormal vectors

Suppose $u_{10}, u_{20}$ are two fixed orthonormal vectors in $\mathbb R^D$. $u_1, u_2$ are also two orthonormal vectors in $\mathbb R^D$ and they are individually uniformly distributed on the unit ...
Landon Carter's user avatar
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Adapting an integration formula for radial functions.

Consider the usual Lebesgue measure and Lebesgue integral over $\mathbb R^n$. An integrable function $f\colon \mathbb R^n \to \mathbb R$ is said to be a radial function if there exists a function $\...
Temirbek Alikhadzhiyev's user avatar
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1 answer
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Confusion on "almost everywhere defined" function in $L^1$ space.

The following are excerpt from folland This proposition shows that for the purposes of integration it makes no difference if we alter functions on null sets. Indeed, one can integrate functions $f$ ...
juekai's user avatar
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2 votes
1 answer
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Showing that $\varphi(x)^p = \sum_{k=2}^\infty \chi_{B_k}(x)$ is locally integrable in $\mathbb R^n$ and satisfies additional property.

Consider the function $$\varphi(x)^p := \sum_{k=2}^\infty \chi_{B_k}(x), \quad \forall x \in \mathbb R^n,$$ where $B_k = B(2^k e_1,1)$ and $1 \leqslant p < \infty$. My goal is to show that this ...
Temirbek Alikhadzhiyev's user avatar
3 votes
1 answer
236 views

Adapting a proof of the non-separability of Morrey Spaces for a different definition.

In the article "Morrey spaces, their duals and preduals", by Marcel Rosenthal and Hans Triebel, for every $1 \leqslant p < \infty$ and $-\frac{n}{p} < r < 0$ the Morrey Spaces are ...
xyz's user avatar
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Simple functions convergence under different topologies

In Serge Lang's Real and Functional Analysis, first part of Lemma 3.1 (p.129) states Let $\{f_n\}$ be a Cauchy sequence of step mappings. Then there exists a subsequence which converges pointwise ...
user760's user avatar
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Essentially bounded functions in $L^{\infty}([0,1],\mathbb{R}^d)$ and their definition

Given a function $f\in L^{\infty}([0,1],\mathbb{R}^d)$. If a consider the following Cauchy problem: $$ \dot{y}(t) = f(t), \quad \text{for almost every } t \in [0,1]. $$ I want to understand, why often ...
hanava331's user avatar
1 vote
1 answer
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$p$-integrability of a certain Bessel function

I read the following statement in a book: $$ \frac{J_{\frac{n-2}{2}}(2\pi r)}{r^{\frac{n-2}{2}}}\text{ is in }L^p\text{ if and only if }p>\frac{2n}{n-1}\text{ ;} $$where $n\geq3$, $r$ is non-...
A. Bond's user avatar
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2 votes
1 answer
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Essentialy bounded functions with compact support are locally integrable and satisfy aditional condition?

Consider arbitrary elements $1 \leqslant p < \infty$ and $0 < \lambda \leqslant n$. Furthermore, during this post I considered the usual Lebesgue measure over $\mathbb R^n$ and I denote the ...
Temirbek Alikhadzhiyev's user avatar
4 votes
0 answers
205 views

Issue with calculating triple Lebesgue integral

I am having issue to understand, why using two approaches I get two different results, i.e. where I am making a mistake: Calculate integral: $$\int_{M}{\sqrt{x^2 + y^2 + z^2}}$$ where: $M := \{(x, y, ...
meerkat's user avatar
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What does "outset" mean in this context?

I'm studying Principles of Mathematical Analysis written by Rudin. I encountered the word "outset" while I was studying Lebesgue integration. The word "outset" appears in page317 ...
jjw's user avatar
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Integrability of a pointwise limit of integrable functions with an extra condition

Assume that $(f_n)$ is a sequence of Lebesgue integrable functions on $(0,1)$ such that \begin{align*} (1) &\lim_{n\rightarrow\infty}\int_0^1f_n \quad \textrm{exists}, \\ (2) &\lim_{n\...
Tony419's user avatar
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Why is Hausdorff Measure used on surfaces integrals?

I have a mild curiosity as to why are we using the Hausdorff measure to define surface integrals (for example the co area formula) and not use instead the Lebesgue measure, or at least on my class and ...
Ramiro genta's user avatar
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1 answer
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Density of Schwartz functions on $H^2(\mathbb{R}^d)$ and the conservation of $L^2$ norm for classical solutions of the Schrodinger equation

We study the Schrodinger equation for a quantum particle in an external potential $V$. We fix an initial wave function $\psi_0 \in L^2(\mathbb{R}^d)$ with $\Vert \psi_0 \Vert_{L^2} =1$ describing the ...
nomadicmathematician's user avatar
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0 answers
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Measurability of the Wiener measure with respect to the starting point

Let $M$ be a Riemannian manifold, and let $C = \{ c : [0,1] \to M \mid c \text{ is continuous}\}$. Endow $C$ with the Wiener measure $\mathbb P_x$ concentrated on the curves $c \in C$ with $c(0) = x \...
Alex M.'s user avatar
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1 answer
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Path integral formalization using measure-based integration and equivalence with limit-based definitions

This question is coming from a computational physicist who is very comfortable with numerical and computational math, but much less comfortable with distributions, measures, and Lebesgue integration. ...
Jonathan Moussa's user avatar
2 votes
1 answer
52 views

How to prove $\int f d\mu=\sup_{n\geq 1}\int f d\mu_n$.

Suppose we have a measurable space $(X, A)$, $\{\mu_n\}_{n\geq1}$ is a sequence of increasing measures on it. (i.e. $\mu_n(E) \leq \mu_{n+1}(E), \forall E \in A$). First prove $\mu(A)= \sup(\mu_n(A))$ ...
Andrew_Ren's user avatar
3 votes
1 answer
135 views

Spectrum of linear operator $Tf(x) = f(x+1) + f(x-1)$ on $L^2(\mathbb{R})$

We are working on the Hilbert space $H = L^2(\mathbb{R})$ and consider the bounded linear operator $T : H \to H$ defined by $(Tf)(x) = f(x+1) + f(x-1)$. What is the spectrum of $T$? What I've tried: ...
Neckverse Herdman's user avatar
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1 answer
77 views

Proving an inequality related to Lebesgue integrals and bounded domains.

Let $\Omega$ be a bounded domain ($|\Omega| < \infty$, where $|\cdot|$ represents the Lebesgue measure), $1 \leqslant p \leqslant q < \infty$, $0 \leqslant \lambda, \mu \leqslant n$ such that $$ ...
xyz's user avatar
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4 votes
2 answers
83 views

$\lim_{{n \to \infty}} \int_{{\frac{1}{n^2}}}^{{n}} (n^2x - 1)e^{-n^2x^2} \,dx$

The limit of the integral is given $\lim_{{n \to \infty}} \int_{{\frac{1}{n^2}}}^{{n}} (n^2x - 1)e^{-n^2x^2} \,dx$ (a) Express the integral in the form $\int_0^\infty f_n(y) \, dy$ where $f_n(y)$ ...
General123's user avatar
1 vote
0 answers
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A bounded measurable function

Let $g:R→R$ be a non-negative integrable function. Let $f:R→R$ be a bounded measurable function satisfying $f(x)>1$ for every $x∈R$. Suppose that $∫_R f^n g≤M$ for every $n∈N$. Show that $g(x)=0$ ...
user1281744's user avatar
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1 answer
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If $\mu$ is a Borel measure on $[0,1]$ and $\int fg = \int f \cdot \int g$ (wrt $\mu$), then $\mu = \delta_a$ (Dirac measure) for some $a \in [0,1]$

A Borel measure is a measure on the Borel $\sigma$-algebra that assigns a finite value to every compact subset. Let $\mu$ be a Borel measure on $[0,1]$ (with the standard topology) and assume that $$\...
Neckverse Herdman's user avatar
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1 answer
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Lebesgue Dominated Convergence Theorem Application

Let $(X_i)_{i\geq 1}$ be a sequence of random variables and $W=|X_1|+\Sigma_{i=1}^∞ |X_{i+1}-X_{i}|$. If $E(W)<\infty$, show that $\lim_{x\to\infty} E(X_n) = E(\lim_{x\to\infty} X_n)$. I'm thinking ...
john22445's user avatar
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1 answer
78 views

Equivalence of Lebesgue-Integral definitions

I am trying to show the equivalence of two definitions of the Lebesgue-integral for functions $f\colon\mathbb{R}^k\to\mathbb{R}$. The first book (Königsberger, Analysis 2, Springer) defines a semi-...
RobRTex's user avatar
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1 answer
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If a function $f : \Omega \to [0,\infty]$ is positive for uncountably many $x \in \Omega$, then counting measure integral on $\Omega$ is infinite

Let $(\Omega, 2^{\Omega}, \mu)$ be the measure space with $\mu$ the counting measure on a set $\Omega \neq \varnothing$ and $f : \Omega \to [0,\infty]$ a function. (All such functions are ...
Neckverse Herdman's user avatar
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1 answer
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Can a sequence of nonintegrable nonnegative measurable functions converge to an integrable function?

Let $X$ be a measure space, and $\{f_n\}$ a decreasing sequence of nonnegative measurable functions $X\to\mathbb{R}$. Is it possible that none of $f_n$ is integrable (i.e., $\int f_n=\infty$ for all $...
ashpool's user avatar
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6 votes
2 answers
144 views

Proving a limit using the dominated converge theorem.

Let $\Omega \subset \mathbb{R}^n$ be an open set and consider the usual Lebesgue space $L^p(\Omega)$. Adicionally, consider the space $$ U(\Omega) = \left\{ f \in L^p(\Omega) \, \colon \, \lim_{r \to ...
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