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
23 views

Hardy-Littlewood-Sobolev Type Prelim Question

The following prelim question bothers me: Question. Let $f\in L^{\frac{3}{2}+h}(\mathbb{R}^{3})\cap L^{\frac{3}{2}-h}(\mathbb{R}^{3})$, for some small $h>0$. Show that $u=f\ast\frac{1}{|x|}\in ...
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1answer
42 views

Show that a linear functional T is bounded if and only if it is Lipschitz

A functional $T$ on a normed linear space $X$ is said to be Lipschitz provided there is a $c \geq 0$ such that $|T(g) - T(h)| \leq c \|g -h\|$ for all $g, h \in X$ The infimum of such c's is called ...
2
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1answer
77 views

Lebesgue measure on a continuous function to prove equality

Let $f(x)$ be a continuous function on [−1, 1]. Show that there exists a constant $c$ such that the Lebesgue measures $\mu (\{x ∈ [−1, 1] : f(x) ≥ c\}) ≥ 1$,$\quad$ $\mu (\{x ∈ [−1, 1] : f(x) ≤ c\}) ≥...
2
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1answer
129 views

Expectation defined as Riemann integral

I have a question related to the expectation of a continuous random variable and its Riemann integral definition. Consider a continuous real-valued random variable $X$ defined on the probability space ...
1
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1answer
58 views

Show that this integral is finite: $ \int_{\mathbb R^3} e^{-\| x\|^2} e^{- a \| x\| \coth (\| x\|) -\| x\| } \, dx $

Haw to prove that the following integral $$ \int_{\mathbb R^3} e^{-\| x\|^2} e^{- a \| x\| \coth (\| x\|) -\| x\| } \, dx $$ is finite ? where $a>0$. thanks you in advance
4
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1answer
82 views

Absolutely continuous function on R

What is the definition of absolute continuity in whole $\mathbb{R}$. I know the definition on an interval $[a, b]$. I have a trouble with understanding the definition of absolute continuity in whole $...
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0answers
31 views

Applicability of monotone convergence theorem, dominated convergence theorem and Fatou's Lemma

In the Monotone Convergence Theorem, the Dominated Convergence Theorem and the Fatou's lemma is having Lebesgue Integrable functions (i.e. functions with finite Lebesgue Integral) a necessary ...
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1answer
47 views

Existence Lebesgue integral and Lebesgue integrability of a function

I have a question related to the existence of Lebesgue Integral. Here in the paragraph "signed function", we read that the Lebesgue integral exists provided that $$(1) \min(\int_{E}f^+d\mu, \int_Rf^{...
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0answers
34 views

Is the supremum of a set finite by definition?

I have a question related to the definition of supremum of a set. Consider a set $A\subseteq \mathbb{R}$. My understanding is that the supremum of $A$ is by definition a finite number. Hence, we have ...
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0answers
20 views

Clarification on the existence of integrals and notation

Consider the random variable $X: \Omega\rightarrow \mathbb{R}^l$ defined on the probability space $(\Omega, \mathcal{A}, P)$, with image $\mathcal{X}\subseteq \mathbb{R}^l$. Consider the measurable ...
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2answers
60 views

Multidimensional Riemann integration and notion of volume or Lebesgue theory and notion of measure

I have finished 9 chapters of "Introduction to Analysis" by Maxwell Rosenlicht (1968). The last chapter treats about "Multiple Integrals". I find the notation a bit complicated. Also, author ...
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1answer
46 views

Markov inequality in real analysis and in probability

The following is the definition of Markov inequality in probability: measure theory: If I want to relate both, according to the definition of expected value: $$\mathbb{E}(|f|)=\int_{-\...
3
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2answers
49 views

Pointwise A.E. Convergence of Convolution

The following is a prelim question, which I can't seem to show under the hypotheses given. Problem. Let $f$ and $g$ be bounded measurable functions on $\mathbb{R}^{n}$. Assume that $g$ is ...
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0answers
50 views

What are some motivations that led to the development of the Lebesgue integral?

There are many kinds of integrals, with the most famous being the Riemann integral which is taught in elementary calculus classes. The motivation behind the Riemann integral is to find the area ...
2
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1answer
95 views

Convergence of the $L^p$ norm to $L^{\infty}$ norm

Let $E \subset \mathbb{R}^n$ measurable. Prove that if there exist $p_0 \geq 1$ such that $f \in L^{p_o}(E) \cap L^{\infty}(E)$, then $f \in L^p(E)$ for all $p \geq p_0$ and $\|f\|_p \rightarrow \|f\|...
2
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2answers
43 views

Integrability of $\frac{1}{|x|^d}$

For $x \in \mathfrak R^d$,why is $\int_\limits{\{x; |x|\geq 1\}} \frac{1}{|x|^d} dx = \infty$ in Lebesgue integral? It's hinted to apply Tonelli Theorem (Fubini Theorem) and use the fact that $\frac ...
5
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4answers
74 views

The infinite sum of integral of positive function is bounded so function tends to 0

Let $f_n(x)$ be positive measurable functions such that $$\sum_{n=1}^\infty \int f_n \lt \infty.$$ Show that $f_n \to 0$ almost everywhere. Attempt: Let $\displaystyle K = \sum_{n=1}^\infty\...
2
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1answer
23 views

derivative of lebesgue integrable function

Suppose we have $f \in L(I)$ and derivative $f'$ exists almost everywhere . It is $f'$ measurable ? I have no idea how to begin to construct the proof .
2
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1answer
42 views

show that $ f_h \in L^1(R) $ and $ \lim_{h \to 0} f_h(x) = f(x) $ in $ L^1(R) $

If $ f \in L^1(R) $ and set $ f_h(x)= \frac{1}{2h} \int_{x-h}^{x+h}f(t)dt, h>0 $ then show that $ f_h \in L^1(R) $ and $ \lim_{h \to 0} f_h(x) = f(x) $ in $ L^1(R) $ To prove f_h is integrable ...
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1answer
42 views

Exploring the total variation of a $C^1$ function

We define the Banach space of functions of bounded variation on $\Omega\subseteq\mathbb{R}^n$ (assume as smooth a domain as we need) as all $u\in L^1(\Omega)$ for which $$\|u\|_{BV}:=\|u\|_1+\int|Du|&...
3
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1answer
71 views

Tao's explanation on how to avoid disjointness for definition of simple function in derivation of Lebesgue Integral

Studying the Lebesgue Integral I am moving back and forth from different books (...I know, bad habit!), and I could not really figure out why sometime, when dealing with the definition of simple ...
2
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1answer
29 views

Convolution as a $L^1$ limit of translates.

I would like what convolution is, as a $L^1$ limit. Namely let $f,g\in L^1(\mathbb{R})$ (with some further conditions). Then what conditions on $f$ and $g$ ensure that $f\ast g$ is the $L^1$ limit for ...
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1answer
153 views

Gap in My Understanding of Measurable Functions as Pointwise Limits of Simple Functions

I've been thinking back to the proof that in $\mathbb{R}$, a measurable function $f:\mathbb{R}\to\mathbb{R}$ is the pointwise limit of increasing simple functions $s_n$. As far as the intuitive ...
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1answer
42 views

Do we have $ \int_{A} \chi_B \mu = \mu(A) $ in a probability space?

Here I have a rather naive question concerning integral representation of probability measures. In general I have problems with it, so here there is a super basic setting: $(X, \Sigma, \mu)$ ...
2
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1answer
27 views

Lebesgue integral in unit circle

I have this doubt that I cannot solve. $\int \limits_{D}\dfrac{|x−1|^a}{|x^2−y^2|^b} \, dx \, dy$ where $D=\{(x,y)\in \mathbb{R}^2: x^2+y^2<1\}$ If I use polar coordinates I cannot solve anything. ...
1
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1answer
51 views

Is $f(x)=\frac{\sqrt{1-x}}{\ln{x}}$ on $[0,1]$ a Lebesgue-integrable function?

I have to prove that $\displaystyle x\mapsto\frac{\sqrt{1-x}}{\ln{x}}$ is Lebesgue-integrable on $[0,1]$. So I try to bound $\displaystyle\left|\frac{\sqrt{1-x}}{\ln{x}}\right|$ with a Lebesgue-...
1
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1answer
53 views

Probability distribution over probability distributions

This may not be a well defined question, but I couldn't find anything about it. Perhaps, I am looking with wrong keywords. Anyway, suppose $X$ is a set, possibly a subset of an Euclidean space. Define ...
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0answers
19 views

Change of variable for non decreasing function

I need help with this problem: Let $g,F \colon \mathbb{R} \to \mathbb{R}$ two continuous and non decreasing functions. Let $\mu_F$ be the Lebesgue-Stieltjes measure associated with $F$. Prove that ...
2
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1answer
31 views

Showing properties of a space using dense subsets (soft)

I'm noticing a lot of times during my functional analysis course, that I'm missing some calculus basics (2 years passed since my last class covering this stuff): Especially when working with Lebesgue- ...
1
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1answer
62 views

Let $f$ be of bounded variation on $[a,b]$, and define $v(x) = f_{[a,x]}$. show $\int_a^b |f'|\leq TV(f).$

Let $f$ be of bounded variation on $[a,b]$, and define $v(x) = TV(f_{[a,x]})$ for all $x \in [a,b]$. show that $|f'| \leq v'$ a.e. on $[a,b]$, and infer from this that $$\int_a^b |f'|\leq TV(f).$$ ...
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0answers
27 views

Taylor series Integral

When can we use Taylor series expansion and write $\int_0^{\infty} \log(f(x+\alpha x)) dx = \int_0^{\infty}\log(f(x)+\sum_{n=1}^{\infty}\frac{f^{n}(x) (\alpha x)^n}{n!}) dx$? I think, first the ...
1
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1answer
63 views

Using Lax-Milgram for linear ODEs

Consider an Sturm-Liouville deferential equation as: $$Lu=(pu')'+qu$$ and differential equation as: $$Lu+f=0$$ where $u(a)=u(b)=0$. We can convert the problem into a Lax-Milgram form for $f\in C[a,b]$ ...
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0answers
44 views

Why this function is not Lebesgue integrable?

Is a function like $H(x) = \left\{ {\begin{array}{*{20}{c}} { 0}&{x < 0}\\ { + \infty }&{x = 0}\\ 1&{x > 0} \end{array}} \right.$ Lebesgue integrable on interval $(-1,1)$? It looks ...
2
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1answer
55 views

Compute $\lim_{n\to\infty }\int_E \sin^n(x)dx$

Let $E$ Lebesgue measurable of finite measure. Compute $$\lim_{n\to\infty }\int_E\sin^n(x)dx.$$ I already have the solution, but I did differently, and I would like to know if it's correct or not. ...
0
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1answer
17 views

Value of the function $\phi=c_1 \chi_{I_1} + c_2 \chi_{I_2}$

Lets say I have the function $\phi=c_1 \chi_{I_1} + c_2 \chi_{I_2}$ When $I_1$ and $I_2$ are disjont, why does $\phi$ take the constant value of $c_1$ at the point of $I_1$ and the constant value of $...
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1answer
54 views

Integral of the characteristic function.

I know $\chi$ is the characteristic function. Why does $\int \chi_I= L(I)$? where L(I) is the length of I.
0
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1answer
23 views

Problem with Lebesgue-integral of measurable set and Lebesgue-integrable function

I am trying to show, that Let $\Omega \subset \mathbb{R}^n$ be measurable and $f\colon \Omega \rightarrow [0,\infty)$ Lebesgue-integrable. Show that: $$\int_\Omega f(x)dx=\int_0^\infty \lambda(\{f&...
3
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1answer
58 views

Eigenvalues of an integral operator

The following operator is defined on $L_2(0,1)$: $$Kf(t)=\int_0^1|s-t|f(s)ds$$ I am wondering how I can calculate the eigenvalues and eigenfunctions of such an operator. I start with $\int_0^1|s-t|f(...
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2answers
27 views

Show that $f_n\to f$ in the norm $L^1(\mathbb{R})$ for $f\in L^1(\mathbb{R})$.

Let $f\in L^1(\mathbb{R})$. Define $$f_n(x)=\begin{cases} f(x) & \text{if }|x|\leq n\\0 & \text{otherwise}\end{cases}.$$ Show that $f_n\to f$ i.n. This seems really obvious, so I'm not sure ...
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1answer
69 views

Show that every Lebesgue integrable function can be approximated in norm and almost everywhere by a sequence of continuous functions.

Show that every Lebesgue integrable function can be approximated in norm and almost everywhere by a sequence of continuous functions. I'm not sure where to even start with this. The question doesn't ...
5
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1answer
128 views

Let $f$ be integrable over $\mathbb{R}$. Show that the following four assertions are equivalent:

Let $f$ be integrable over $\mathbb{R}$. Show that the following four assertions are equivalent: $f = 0$ a.e. on $\mathbb{R}$. $\int_{\mathbb{R}}fg=0$ for every bounded measurable ...
2
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1answer
39 views

Prove that $f(x,y)=\frac{xy}{x^2+y^2}$ is not Lebesgue integrable on $A = [-1,1]\times [-1,1]$

Prove that $f(x,y)=\frac{xy}{x^2+y^2}$ is not Lebesgue integrable on $A = [-1,1]\times [-1,1]$ To my knowledge I need to use Fubini's theorem. But this doesn't work because the integration would be ...
0
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2answers
46 views

Show that the characteristic function of $\mathbb{Q}$ is Lebesgue integrable.

Show that the characteristic function of $\mathbb{Q}$ is Lebesgue integrable. I've shown that it isn't Reimann integrable, but I'm stuck on showing it is Lebesgue integrable. Any help would be much ...
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1answer
45 views

Prove a function to be analytic by dominated convergence theorem

Given $f \in L^1$, prove that $$F(z) = \frac{1}{2\pi i} \int^\infty_{-\infty} \frac{f(t)}{t-z}\,dt$$ is an analytic function and $$F'(z)=\frac{1}{2\pi i} \int^\infty_{-\infty} \frac{f(t)}{(t-z)^2}\...
1
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1answer
24 views

A bounded family of functions in $L^p[E]$, where E is a measurable set, is uniformly integrable.

A corollary in Royden & Fitzpatrick's Real Analysis (chapter 7 section 2) reads: Let $E$ a measurable set, and $1<p<\infty$. Suppose $F$ is a family of functions in $L^p(E)$ that is bounded ...
2
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1answer
71 views

Lebesgue monotone convergence theorem

I have a doubt regarding the Lebesgue monotone convergence theorem. The version that I know is the following from Wikipedia, requiring, in particular, $\{f_k(x)\}$ monotone increasing and $f_k(x)\...
1
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1answer
20 views

Prove sum is commutative with measure theory result

Given a measured space $(X,\mathcal{A},\mu)$, consider the following function $f:X\rightarrow \mathbb{R}^+$ defined by $$ f(x)=\sum_{n=0}^\infty f_n(x), $$ where $f_n(x)$ are positive measurable ...
1
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0answers
51 views

$L^p$ space and continuous injection

Let $1\leq p < r < q \leq \infty$ and $E\in \mathbb{R}$. Define $$A = L^p(E) + L^q(E) = \{f=g+h:g\in L^p(E), h\in L^q(E) \}$$ and $$\|f\|_A = \inf_{f=g+h} \|g\|_p+\|h\|_q$$ where the infimum ...
1
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1answer
61 views

Prove $\int\limits_{]0,\infty[}\frac{\ln{x}}{x^2-1} d\lambda_1(x)=\frac{\pi^2}{4}$

I try to prove the following statement: $$\int\limits_{]0,\infty[}\frac{\ln{x}}{x^2-1} d\lambda_1(x)=\frac{\pi^2}{4}$$ There is also a clue: $$ \frac{1}{(1+y)(1+x^2y)}=\frac{1}{x^2-1}\left(\frac{x^2}{...
1
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1answer
66 views

Is it always true that the Lebesgue integral of a continuous function is equal to the Riemann integral (even if they are both unbounded)?

Let's assume that $f\colon\mathbb R\to\mathbb R$ is continuous and hence Lebesgue measurable. Then, the Lebesgue integral $\int_{(0,\infty)}f(x)\,d\lambda(x)$ makes sense (but of course can be equal ...