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.

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Images of intersections and Lebesgue integrals

How does the image not commuting with intersections, instead being just a subset of it $$f(U \cap V) \subset f(U) \cap f(V),$$ tell us that the Lebesgue integral is better than the Riemann integral? ...
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1answer
39 views

Absolute continuity and derivatives of integrals

I am preparing for a comprehensive at the end of the month, so I would appreciate any input I could get on this solution. I am pretty confident if the first part, but I think the second answer could ...
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43 views

Introduction of Lebesgue Theory

While reading Real Analysis- Stein, Shakarchi, I came up with following questions, some of which are in the "Introduction" of the book. The purpose of a series of questions is "to get actual ...
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2answers
105 views

Physical interpretation of L1 Norm and L2 Norm

In signal analysis, students have no qualms about associating the L2 norm of a square integrable function f(t) as the energy associated with that signal. A good understanding of whether a function ...
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1answer
33 views

Calculate Radon-Nikodym derivative in a point when it is continuous in that point

I can't solve the following exercise, even if I find it quite intuitive. Let $\nu, \mu$ be Radon measures on a metric space $(X,d)$. Suppose that: 1) $w\in L^1(X,\mu), w\geq 0$ $\mu$ a.e.; 2) $w$ is ...
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1answer
12 views

can representatives of an equivalence class of L_1(r^d)be nonmeasurable

Motivation: The composition of 2 Lebesgue measurable functions need not be measurable. This problem can be dealt with in a case by case basis( like with convolutions). Or as Big Rudin does, apply ...
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4answers
91 views

Can someone provide an example of Lebesgue integration on ordinary functions?

I'm trying to understand Lebesgue integral and integration but I'm completely mired by many author's motivation of this subject as only a tool for pathological functions such as the Dirichlet ...
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34 views

Prove something is a signed measure

Given a measure space $(X,\mathcal{M},\mu)$ and a measurable function $f:X\rightarrow \overline{\mathbb{R}}$ such that at least one of $f^+$ or $f^-$ is integrable, show that ...
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18 views

Non Borel Spaces: Gauge Integral

Is there a gauge integral over non Borel spaces? (My interest lies in finite measure spaces.) It seems as the definition of the gauge integral crucially depends on the existence of open sets for a ...
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0answers
87 views

Riemann implies Lebesgue integrablility on $\mathbb{R}^n$, prove $f(x)$ continuous at x where $g(x)=G(x)$

Let $f:[a_1,b_1]\times \cdots \times[a_n,b_n] \rightarrow \mathbb{R}$ be Riemann integrable. Prove that is $f$ Lebesgue integrable. Proof: $$Q:= [a_1,b_1]\times \cdots \times [a_n,b_n].$$ For simple ...
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1answer
37 views

A tricky integral with vanishing domain

I would love to have the following result, however I got no clue if it is even true! Let $B_n:=\{y:\varepsilon_n<|y|\leq\tilde{\varepsilon}_n\}$ for some sequences ...
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64 views

Why is the value assigned to a gauge integral well defined (unique)?

Why is the value assigned to a gauge integral well defined (unique)? If we would have given a net (so an underlying order that happens to be directed), then the limit would be unique given a ...
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2answers
134 views

The set $E= \{x\in [0,1]: \sum_{j=1}^\infty t^j|x−q_j|^{-r} <\infty\}$ does not contain all irrational numbers in $[0,1]$

Let $q_1,q_2,q_3,...$ be an enumeration of $\mathbb{Q}\cap[0,1]$ and let $r,t \in (0,1).$ Consider the set $$E= \{x\in [0,1]: \sum_{j=1}^\infty t^j|x−q_j|^{-r} <\infty\} $$ (a) Show that $E\neq ...
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2answers
82 views

How to prove indicator function, step function, and then for sequences of step functions?

I try to proof a claim. It should be done first for indicator functions, then for step functions and finally for limits of increasing sequences of step functions. I'm not sure if I'm doing it right. ...
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2answers
153 views

Show that $\int_{\mathbb{R}^n}f_1f_2 …f_n dx_1 …dx_n ≤ (I_1 …I_n)^{1/(n−1)}.$

For $\quad k = 1,2,...n,\quad$ let $\quad\mathbb{R}^k = \mathbb{R},\quad f_k(x_1,...,x_{k−1},x_{k+1},\ldots,x_n)\quad$ be a nonnegative measurable function on $\quad\mathbb{R}_1\times\ldots\times ...
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3answers
94 views

Is Dirac's delta function well-defined at Lebesgue points?

Usually in textbooks, $$\int_{\mathbb{R}^d} \delta(\mathbf{x}-\mathbf{y})f(\mathbf{x}) = f(\mathbf{y})$$ holds given $f$ is continuous. On the other hand, the definition of Lebesugue point ...
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42 views

Passing of the limit for Lebesgue Integral (Proof Verification)

Let $f_n\in L^1(0,1)$ and $C>0$ be such that $f_n \geq 0, f_n \rightarrow 0$ a.e., and $$\int_0^1 \max\{f_1, ..., f_n\} dx \leq C \quad \text{ for every } n.$$ Prove that $f_n \rightarrow 0$ in ...
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0answers
25 views

Problem involving decomposition of measures

Let $\mu$ be a signed measure. We wish to prove that $$\left| \int{f} \> d\mu \right| \leq \int{|f|} \> d|\mu|.$$ (We are given the following defintion: $\int{f} \> d\mu = \int{f} \> ...
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3answers
80 views

Lebesgue integral of $\chi_{\mathbb{Q}}: \mathbb{R} \rightarrow \mathbb{R}$

Suppose $(X, \mathfrak{A}, \mu)$ is a measure space. Let $\phi$ be a simple function with canonical representation $\sum^{k}_{n=1} a_{n} \chi_{E_{n}}$. I know we define the Lebesgue integral of $\phi$ ...
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51 views

About the $\lim_{n \to+\infty} \frac{1}{n}\int_0^1 \log(1+e^{nf(x)})\,dx$ (Rudin's exercise)

Problem (Rudin, R&CA chapter 2, no. 25) (i) Find the smallest positive constant $c$ such that $$ \log(1+e^t) \le c+t , \qquad t \in (0,+\infty). $$ (ii) Does $$ \lim_{n ...
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1answer
36 views

Finite Measure Space: Integral Closure = Bochner Integral

I can't sleep for so long time as the integral gives me headaches. I was looking for so many approaches. Now another one. Hope this works... Let $\Omega$ be a finite measure space and $E$ a Banach ...
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1answer
36 views

How do the inner products on $L^2$ look like?

I was wondering whether all scalar products in $(L^2[0,1],\lambda)$ are given by $\langle f,g \rangle := \int f(x)g(x) \cdot w(x) d\lambda(x)$? If this is true, what are the exact conditions that we ...
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0answers
35 views

Not lebesgue integrable function?

I want to consider the function $f:[-1,1]\times [-1,1]\rightarrow \mathbb R:f(x,y)= \begin{cases} \frac{xy}{(x^2+y^2)^2} & (x,y) \neq (0,0) \\ 0 & (x,y) = (0,0) \end{cases} $ And I have ...
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37 views

Prove that the function $g(·)$ is twice continuously differentiable and that $g′′(α) ≥ 0$ for all $α ∈ \mathbb{R}$, i.e.

Let $f$ be a real Lebesgue measurable function on the interval $[0, 1]$ such that $∥f∥∞ < ∞.$ For $α ∈ \mathbb{R}$ define a function $g(α)$ by $g(α) = \log \int_0^1\exp[αf(x)] dx .$ (a) Prove that ...
3
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1answer
43 views

Inequality involving Jensen (Rudin's exercise)

Exercise (Rudin, R&CA, no. 3.25). Suppose $\mu$ is a positive measure on the space $X$ and let $f \colon X \to (0,+\infty)$ be such that $\int_X f \, d\mu=1$. Then for every $E \subset X$ ...
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1answer
53 views

Find $\lim_{n \rightarrow \infty}$$\int_0^n(1 + \frac{-x}{n})^n\cos(\frac{x}{\sqrt{n}})e^{x/2}dx$

Find $\lim_{n \rightarrow \infty}$$\int_0^n(1 + \frac{-x}{n})^n\cos(\frac{x}{\sqrt{n}})e^{x/2}dx$ I want to use dominated convergence theorem obviously. However, not sure how to dominate it. ...
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43 views

Prove that $f\ast g$ is defined a.e., integrable, and such that $∥f\ast g∥_1 ≤ ∥f∥_1 · ∥g∥_1$

Let $f,g : \mathbb{R} → \mathbb{R}$ be $L_1$-functions. Set $h(x) = \int_\mathbb{R}f (x − y)g(y) \, dm(y).$ Prove that $h(x)$ is defined a.e., $h ∈ L_1(\mathbb{R})$ and $∥h∥_1 ≤ ∥f∥_1 · ∥g∥_1.$ So I ...
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1answer
84 views

Integration by substitution and separation of variables.

Let's say I want to integrate over a sphere $S^2$. Take $f \in (L^1(S^2),dS)$, then we have that $$\int_{S^2} |f| dS = \int_{S^2} |f| \sin^2 (\theta) d \theta d \phi < \infty,$$ right? Now, ...
3
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1answer
50 views

Let $(X,μ)$ be a measure space. Find a necessary and sufficient condition on $(X,μ)$ that $L_q(E) ⊂ L_p(E)$ for all $1 ≤ p < q ≤ ∞.$

Let $(X,μ)$ be a measure space. Find a necessary and sufficient condition on $(X,μ)$ that $L_q(E) ⊂ L_p(E)$ for all $1 ≤ p < q ≤ ∞.$ I want to say that the condition is that $E$ is finite. This ...
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39 views

Show that $\int_E f (x, y) dx$ is differentiable with respect to $y$ and $\frac{d}{dy}\int_E f(x,y)dx=\int_E \frac{d}{dy}f(x,y)dx.$

Assume that $f = f(x,y)$ is a function defined on $E × (a,b).$ For each fixed $y ∈ (a,b),$ $f$ is integrable with respect to $x$ on $E$, and for each fixed $x ∈ E$, $f$ is differentiable with respect ...
2
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1answer
47 views

Are these two expression square integrable?

I have two expressions (let's call them functions $f,g$) on $[0,1]$, where I want to find out whether they are square-integrable or better: for which $m \in \mathbb{Z}$ they are square-integrable ( ...
3
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1answer
30 views

Prove this function is absolute continuous and Lipschitz of order $\alpha$

Let $1≤p≤\infty$ and $ f \in L^{p}(a,b)$ such as there is a function $g \in L^{p}(a,b)$ that for all $\phi \in C^{1}(a,b)$ (and continuos in $[a,b]$) with $\phi(a)=\phi(b)=0$ we have: $\int_{a}^{b} ...
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2answers
40 views

For $1 \leq r < p < \infty$ prove the continuous injection of $L^p([0, 1])$ into $L^r([0, 1])$.

For $1 \leq r < p < \infty$ prove the continuous injection of $L^p([0, 1])$ into $L^r([0, 1])$. I am having a hard time starting. Any suggestions. I tried a straight forward approach. That ...
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1answer
39 views

Spherical coordinates and Lebesgue integral

I found a textbook that says that $r \in [0,\infty), \theta \in [0,\pi], \phi \in [-\pi,\pi]$ are the spherical coordinates. Then he goes on with: This corresponds to a unitary map $L^2(\mathbb{R}^3) ...
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1answer
34 views

What is the definition of this set of absolutely continuous function

I know that $$AC(a,b):=\left\{f \in C(a,b)|f(x) = f(c)+\int_c^x g(t) d \lambda(t),c \in (a,b), g \in L^1_{\text{loc}}(a,b)\right\}$$ $$AC[a,b]:=\left\{f \in C[a,b]|f(x) = f(c)+\int_a^x g(t) d ...
2
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1answer
50 views

Find a non-negative function on [0,1] such that $t\cdot m(\{x:f(x) \geq t\}) \to 0$ that is not Lebesgue Integrable

Problem: Find a non-negative function $f$ on $[0,1]$ such that $$\lim_{t\to\infty} t\cdot m(\{x : f(x) \geq t\}) = 0,$$ but $f$ is not integrable, where $m$ is Lebesgue measure. My Attempt: Let ...
5
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1answer
46 views

Let $f_n:\mathbb{R}\rightarrow [0, 1]$ be functions such that $\sup_{x \in \mathbb{R}}f_n(x) = 1/n$ and $||f||_1 = 1$.

Let $f_n:\mathbb{R}\rightarrow [0, 1]$ be functions such that $\sup_{x \in \mathbb{R}}f_n(x) = 1/n$ and $||f_n||_1 = 1$. Set $F(x) = \sup_{n \in \mathbb{N}}f_n(x)$. Prove that $\int_\mathbb{R}F(x)dx ...
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0answers
39 views

Absolute continuous function and Jensen inequality

Let $f:[0,1]\rightarrow \mathbb{R}$ be an absolute continuous function, such as $f' \in L^4(0,1)$ and $f(0)=0$. Find $r<0$, such as, for all $\alpha \geq r,$ $lim_{x\rightarrow 0^{+}} ...
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1answer
40 views

A Fubini-Tonelli's Theorem Problem

Let $E \subseteq \mathbb{R^n}$ a measurable set, such as for almost every $x \in \mathbb{R^n}$ we have $|E \triangle (E+x)|=0$ (Where $\triangle$ means simetric difference between two sets and ...
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1answer
44 views

Approximation functions in $L^{1}$ by indicator functions of dyadic cubes

Let $\mu$ be a finite positive regular Borel measure on $\mathbb{R}^{d}$ and let $S$ be the family of finite unions of squares of the form $\{a_{1}2^{n} \leq x_{1} \leq (a_{1} + 1)2^{n}, \ldots, ...
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1answer
32 views

$L^\infty(\Omega)$ space

Consider Lebesgue spaces $L^p(\Omega)$, $\Omega$ is a bounded domain. Let $f \in L^p(\Omega)$ for all $p$. Is it true that $f \in L^\infty(\Omega)$?
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1answer
51 views

Changing the order of integration (for Lebesgue-Stieltjes integral and Riemann integral)

Do the Lebesgue-Stieltjes integral and the Riemann integral have the same rules about the change of order of integration? I mean I know how to deal with Riemann integral, but I'm not sure if I can ...
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0answers
53 views

Why $\int f_0 \mbox{d}(F_0+F_1)=\int f_0 \mbox{d}F_0+\int f_0 \mbox{d} F_1=1$ should be true?

A measure $\mu$ dominates another measure $\nu$ whenever $\mu=0$ implies $\nu=0$. If I would like to take the integral of a measurable function $f_0$, say the density function of the probability ...
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2answers
62 views

$f$ unbounded in $\mathbb{R}$ implies it cannot be in $L^2(\mathbb{R})$.

Intuitively , I feel this must be true. I'm looking for a rigorous proof. So,I would like to confirm that there are no counter examples to begin with.
5
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1answer
67 views

$\int_0^1f(x)dx = 2, \int_0^1g(x)dx = 1, \text{and} \int_0^1[f(x)]^2 dx ≤ C$ for some constant $C > 4.$

Suppose $f$ and $g$ are nonnegative measurable functions on the interval $[0,1],$ with the properties $$\int_0^1 f(x)\,dx = 2, \int_0^1g(x)\,dx = 1, \text{ and }\int_0^1[f(x)]^2 dx \le C$$ for some ...
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2answers
28 views

Show that $\int_X gdν=\int_X gfdμ$ for all $g∈L_1(ν).$

Let $μ$ and $ν$ be finite (positive) measures on a measurable space $(X, M),$ and suppose that $ν(E)=\int_E fdμ$, for all $E∈M,$ $E$ where $f$ is some function in $L_1(μ).$ Show that $\int_X ...
2
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0answers
26 views

Integrals depending upon a parameter

There was an exercise, in my professor's book, asking to prove the continuity of an integral depending upon a parameter. Namely, the hypothesis were: Let $D$ be a measurable subset of $\mathbb{R}^n$, ...
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1answer
53 views

Integration: Countable Additive Measure?

When considering Bochner's theory of integration one notices that having a countable additive measure rather than merely additive measure is not important, or do I miss something?
2
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2answers
75 views

Proving Lebesgue integration result

I have a Lebesgue integration question and a proposed proof. Please advise. Let $\Omega \subset \mathbb{R}^{n}$(denote the boundary as $\partial \Omega$) and consider $$\int_{\partial \Omega} vf ...
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1answer
37 views

Uniform and integral limit

Let $f_n(x)=n(\sin x)^n \cos x$. Show that the sequence of functions $f_n$ converges to $0$ uniformly on any interval of the form $[0,a]$ where $a<\pi /2$. Show that, for any continuous function ...