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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-3
votes
1answer
22 views

Use monotone convergence theorem to find $\int_{(0,1)} gdm$ where $g(x)=x^{-\frac{1}{2}}$. [closed]

Use monotone convergence theorem to find $\int_{(0,1)} gdm$ where $g(x)=x^{-\frac{1}{2}}$. Any idea which sequence of functions should I use? Edit: $m$ is standard Lebesgue measure on $\mathbb{R}$. ...
6
votes
2answers
264 views

Counterexample of the almost-inverse of the Fundamnetal Theorem of Calculus(Lebesgue).

Can anyone give me a counterexample to the following statement: Suppose $F \colon [0,1] \to \mathbb{R}$ is continuous and differentiable almost everywhere, then $F(b)-F(a)=\int_a^b F'(t)\, ...
5
votes
1answer
93 views

Product of $L^{1}$ Function and Exponentially Integrable Function

Problem. Let $g\geq 0$ be in $L^{1}[0,1]$ and suppose that $\int gfdx<A$ whenever $\int e^{f}dx\leq 1$. What can one say about $|\{g>\lambda\}|$ for $\lambda\gg 1$. Is $g\in ...
0
votes
1answer
28 views

Use monotone convergence theorem to show that $f(x)=\frac{1}{x}$ is not summable on $(0,1)$

Use monotone convergence theorem to show that $f(x)=\frac{1}{x}$ is not summable on $(0,1)$. The hint was to consider the sequence $f_n(x)=\max \{n, f(x)\}$, $x\in(0,1)$. It is clear that $0\le f_1 ...
0
votes
1answer
24 views

Understanding the proof for Lebesgue integrability criterion

I found a somewhat rigorous proof of Lebesgue's integrability criterion, where $\Delta x_k = x_k - x_{k-1}$, and $$M_k = \sup_{x \in [x_{k-1},x_k]} f(x),m_k = \inf_{x \in [x_{k-1},x_k]} f(x) $$ ...
2
votes
2answers
49 views

Using Lebesgue's dominated convergence theorem to show a function is continuous.

I have a function $U(t)=\int_\mathbb{R} u(x) \cos(xt)dx$ and I am trying to use Lebesgue's dominated convergence theorem to show $U(t)$ is continuous for all $t \in \mathbb{R}$ This is the proof. ...
3
votes
1answer
21 views

Show that $f$ is summable on $A$ and $\lim_{n\rightarrow \infty} \int_A f_n dm=\int_A f dm$

Show that if $f_n$ is summable on a bounded measurable set $A$ for $n=1,2,\ldots$ and if $f_n$ converges uniformly to $f$ on $A$ then $f$ is summable on $A$ and $\lim_{n\rightarrow \infty} \int_A f_n ...
1
vote
2answers
75 views

Proving $f$ is $L^1$ implies convergence of $\sum_{k \in \mathbb{Z}}{ 2^k*m(|f| > 2^k)}$

I need help proving that if f is Lebesgue integrable then $$\sum_{k \in \mathbb{Z}}{ 2^k*m(|f| > 2^k)}$$ converges. I proved the opposite direction but I don't know how to put any bounds on each ...
3
votes
1answer
53 views

Definition of integral in the context of measure theory

Let $\left( X, \mathcal{F}, \mu \right)$ be a measure space. I want to make sense to the integral $$\int_{X}f(x)d\mu (x).$$ It's easy to give meaning to this when $f$ is a simple function. But if $f$ ...
4
votes
1answer
103 views

Radon measure determined by the intersection of half lines in the plane

Consider a vector $r$ in the euclidian plane $\mathbb R^2$ and two unit vectors $u,v\in\mathbb U$ ($\mathbb U$ is the unit circle). Let $s>0$ be a real number. I am looking for an expression of the ...
4
votes
1answer
45 views

Weak Limit of Measures Mutually Singular wrt Lebesgue Measure

I'm stuck on the following qual problem: Let $\{h_{n}\}$ be a sequence of positive continuous functions on the unit cube $Q$ in $\mathbb{R}^{d}$ satisfying the following conditions: ...
2
votes
2answers
41 views

Function Riemann integrable that are not Lebesgue integrable.

I had an exam today and one of my question was: Give a function $f$ that is Riemann integrable but not Lebesgue integrable How is it possible ? I always thought that Riemann $\implies $ Lebesgue, ...
1
vote
1answer
22 views

Show $\lim_{n\rightarrow\infty} \int_{E_n}gdm=0\mbox{.}$

Show that if $g$ is summable on $A$ and $A\supset E_1 \supset E_2,\ldots$ and $\cap_{n=1}^{\infty} E_n= \emptyset$ then $$\lim_{n\rightarrow\infty} \int_{E_n}gdm=0\mbox{.}$$ Proof: Assume the ...
0
votes
1answer
23 views

Prove that if $f$ is measurable on a measurable and bounded set $A$ and $\int_B f dm=0$ for each measurable $B\subset A$ then $f=0$ a.e.

Prove that if $f$ is measurable on a measurable and bounded set $A\subset\mathbb{R}$ and $\int_B f dm=0$ for each measurable $B\subset A$ then $f=0$ a.e. WLOG suppose that $S=\{x\in A : ...
0
votes
0answers
23 views

Definition of Lebesgue integral via Lebesgue sums?

In "Introduction to Lebesgue Theory and Fourier Series" I have encountered a bit unusual definition of Lebesgue integral (via the limit of so called Lebesgue sums which are analogous to Riemann sums ...
5
votes
0answers
37 views

Show that if $f$ is measurable then $\{x\in A:c=f(x)\}$ is measurable for each $c$.

Let $A$ be a bounded measurable subset of $\mathbb{R}$. Show that if $f:A\rightarrow \mathbb{R}$ is measurable then $\{x\in A:c=f(x)\}$ is measurable for each $c$. Choose real $c$. Since $f$ is ...
5
votes
1answer
27 views

Proving that the Lebesgue integral over a measurable function $f$ is equal to the area/volume below the graph of $f$

Given a Borel set $A \subseteq \mathbb{R}^d, d ≥ 1$ and a measurable function $f: A \to [0, \infty)$, I want to consider the set: $$E = \{(x, y) \in \mathbb{R}^{d+1}: x \in A, 0 ≤ y ≤ f(x)\} ...
1
vote
0answers
40 views

Proof of Lusin theorem

Theorem (Lusin): Let $f$ measurable and finite valued on $E$ a set of finite measure. For all $\varepsilon>0$ there is a closed set $F_\varepsilon\subset E$ s.t. $f|_{F_{\varepsilon}}$ is ...
2
votes
0answers
29 views

Integration by parts for Dirac measure

We know that Dirac measure is defined by $$\delta_x(A)= \begin{cases} 1 &\text{if $x \in A$}\\ 0 &\text{if $x \notin A$}\\ \end{cases}$$ We know that $\int_a^b f(y) \, d\delta_x(y)=f(x),$ if ...
2
votes
1answer
77 views

Lebesgue integral, Cavalieri's principle

Merry Christmas, Can you prove my answer: Let $B_r^n (p):=\{x\in\mathbb R^n \mid |x-p|\le r\}$ be the Ball with radius $r\in\mathbb R_{+}$ and origin $p\in\mathbb R$ and dimension $n\in\mathbb N$. ...
5
votes
1answer
30 views

Dilations of Integrable Function Converge to Zero Almost Everywhere

Suppose $f:[0,\infty)\rightarrow [0,\infty)$ is integrable. Set $f_{n}(x):=f(nx)$. I want to show that $f_{n}(x)\rightarrow 0$ almost everywhere or equivalently, the set $$\{x : ...
1
vote
1answer
16 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 ...
0
votes
1answer
37 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
votes
1answer
62 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
votes
1answer
111 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
vote
1answer
57 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
3
votes
1answer
62 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 ...
1
vote
0answers
27 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 ...
0
votes
1answer
35 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, ...
0
votes
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 ...
0
votes
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 ...
2
votes
2answers
38 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 ...
1
vote
0answers
33 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: ...
3
votes
2answers
43 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 ...
0
votes
0answers
46 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
votes
1answer
57 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 ...
2
votes
2answers
42 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
votes
4answers
60 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 = ...
2
votes
1answer
22 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 .
1
vote
1answer
40 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 ...
1
vote
1answer
36 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 ...
3
votes
1answer
61 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
votes
1answer
26 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 ...
1
vote
1answer
149 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 ...
1
vote
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
votes
1answer
23 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
vote
1answer
44 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 ...
1
vote
1answer
50 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 ...
0
votes
0answers
12 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 ...
1
vote
1answer
24 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- ...