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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1answer
35 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 ...
3
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0answers
39 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 ...
4
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2answers
106 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 ...
0
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2answers
70 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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0answers
49 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 ...
3
votes
3answers
77 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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0answers
37 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 ...
2
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0answers
23 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} \> ...
1
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3answers
74 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 $\phi$ ...
6
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1answer
46 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 ...
1
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1answer
31 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
34 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 ...
1
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0answers
29 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 ...
0
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0answers
36 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 ...
2
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1answer
35 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 $E \subset X$ ...
0
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1answer
46 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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0answers
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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0answers
64 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
39 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 ...
2
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0answers
35 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
votes
1answer
45 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
votes
1answer
24 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} ...
3
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2answers
35 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 ...
2
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1answer
34 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) ...
2
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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
votes
1answer
45 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
41 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 ...
0
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0answers
35 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^{+}} ...
0
votes
1answer
32 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 ...
1
vote
1answer
36 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, ...
0
votes
1answer
30 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 $f \in L^\infty(\Omega)$?
2
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1answer
39 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
52 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 ...
1
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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
63 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 ...
0
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2answers
26 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 ...
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0answers
21 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 $\mathbb{R}^n$, ...
0
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1answer
46 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
73 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 ...
1
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1answer
34 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 ...
7
votes
2answers
54 views

Integral limit of $\sin(x/n)f(x)$

For any $f\in L^1[0,\pi]$, evaluate $n\to \infty \int^\pi_0 n$sin$(x/n)f(x)dx$ My idea is, $n$sin$(x/n)f(x)\to xf(x)$ and it seems that it is increasing sequence. I am not able to show it is ...
3
votes
1answer
41 views

Finding a dominating function for this sequence of functions

Problem: Find the limit $$\lim_{n\to\infty} \int_0^n \left( 1 + \frac{x}{n}\right )^{-n} \log(2 + \cos(x/n))dx$$ and justify your reasoning. My Solution: Let $f_n = \left( 1 + \frac{x}{n}\right ...
4
votes
1answer
58 views

Functions with every point being a Lebesgue point

For a locally integrable function $f$ a point $x$ is a Lebesgue point if the integral averages of deviations from $f(x)$ over balls centered at $x$ converge to $0$ as the balls shrink to the point. ...
4
votes
7answers
292 views
+50

Assume that $ f ∈ L([a, b])$ and $\int x^nf(x)dx=0$ for $n=0,1,2…$.

Assume that $ f ∈ L([a, b])$ and $\int x^nf(x)dx=0$ for $n=0,1,2...$. Prove $f=0$ a.e. Since there exist polynomials going to f almost everywhere all I would need to do is bring the limit in to ...
-2
votes
2answers
81 views

Group of Unitaries: Strong Continuity

Let $\mathcal{L}^2(\mathbb{R})$ be the the Hilbert space of square integrable functions, shortly $\mathcal{L}^2$. Consider the group of unitaries: $$U:\mathbb{R}\to ...
0
votes
3answers
57 views

If $f:[0,1]\to\mathbb{R}$ is an integrable function, prove the following:

If $f:[0,1]\to\mathbb{R}$ is an integrable function, prove the following: $\displaystyle\lim_{h\to0}\int_{[0,1]}\frac{|1+h\cdot f(t)|-1}{h}dm(t)=\int_{[0,1]}f(t)dm(t)$ I don't even know where to get ...
0
votes
1answer
46 views

Determining whether an uncountable set of integral equations yield a unique solution

I am interested in the set of numbers $\alpha>0$ for which there exists a function $g:\mathbb{R}\to[0,1]$ satisfying $$ \forall r\in \mathbb{R} \qquad f(r) = \int\limits_\mathbb{R}\! g(\alpha ...
1
vote
2answers
40 views

The applicability of the Dominated Convergence theorem on the real line

Let $f_n(x)=\frac{1}{n}\chi_{[0,n]}(x)$, $x\in\mathbb{R}$, $n\in\mathbb{N}$ and $\chi$ is the characteristic/indicator function. Now it is clear that $f_n\rightarrow 0$, but in the text I am using it ...
1
vote
1answer
36 views

Proposed proof Lebesgue integration question

I just want to confirm the following proof: Consider a function $u: \Omega \rightarrow \mathbb{R}$ where $\Omega \subset \mathbb{R}^{n}$ and $u \in C^{2}(\bar{\Omega})$. Let $a_{jk}$ be smooth ...
2
votes
2answers
39 views

For a real valued function $f(x,y)$ on $\mathbb{R}^2$ which is in $L_2$, show that $f(x+ε,y+ε) → f(x,y)$ in $L_2$ when $ε → 0.$ [duplicate]

For a real valued function $f(x,y)$ on $\mathbb{R}^2$ which is in $L_2$, show that $f(x+ε,y+ε) → f(x,y)$ in $L_2$ when $ε → 0.$ Not sure how to go about this problem. I tried Fubini. But that ...