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.

learn more… | top users | synonyms

0
votes
2answers
29 views

Showing that the L1 norm of a given sequence of functions diverges

For $n=1,2,3,\ldots$ define $f_n:\mathbb R\to\mathbb R$ by $$f_n(x) = \frac{\sin(x)\sin(nx)}{x^2}.$$ Then certainly each $f_n$ is integrable on the real line. However, I have to show that the $L^1$ ...
0
votes
1answer
34 views

Show that $g_n$ converges in $L^1(\mathbb{R})$ to some $F$ and find $F$.

My question is : Let $f\in$$L^1(\mathbb{R})$, and define $g_n(x)$=$\frac{1}{n}\sum_{k=1}^{n}f(x+\frac{k}{n})$. Show that $g_n$ converges in $L^1(\mathbb{R})$ to some $F$ and find $F$. First, I ...
0
votes
0answers
23 views

Fubini's Theorem and Toneli's theorem

I'm having a problem in proving: $\int_B \frac{dx}{\mid x \mid^a } < \infty \Leftrightarrow a < n$ , $\int_{B^c} \frac{dx}{\mid x \mid^a } < \infty \Leftrightarrow a > n$ where ...
2
votes
2answers
59 views

Convergence in $L^p$ spaces.

Prove that for all integrable functions $g_n, g$, we have the implication $\|g_n-g\|_1\to 0\Rightarrow \|g_n\|_1\to \|g\|_1$ as $n\to \infty$. Is the converse true? It seems like $|g_n-g|_1 \to 0$ ...
3
votes
2answers
63 views

If $f_n (x)=\frac{n \sin x}{x (1+n^2 x^2)}$ then evaluate limit of integration $f_n(x)$ over $0 \to 1$ as $n \to \infty$

In this problem, I tried to dominated convergence theorem but I couldn't get any dominated function. How to find limit of this integration? Any hints or comments are welcomed.
2
votes
1answer
38 views

Is $\partial (A\times B)$ jordan measurable when both of $A$ and $B$ are jordan measurable?

If $A\subseteq \mathbb{R}^{n} $ is Jordan measurable, $B\subseteq \mathbb{R}^{m} $ is Jordan measurable, then $A \times B \subseteq \mathbb{R}^{n+m}$ is Jordan measurable? We have $$\partial ...
2
votes
1answer
47 views

How to prove or disprove for the statement?

Let $f$ be a measurable function on $\mathbb{R}$ and $p\in[1,\infty)$. By Lebesgue Dominated Convergence Theorem, we know the statement " If $f\in$$L^p(\mathbb{R})$, then ...
2
votes
1answer
39 views

Is the limit equal to zero?

I know the statement " If $f\in$$L^1(\mathbb{R})$ is uniformly continuous, then $\lim_{|x|\to\infty}|f(x)|=0$ " is true. How about $f\in$$L^1(\mathbb{R})$ is continuous but not uniformly continuous ? ...
0
votes
0answers
29 views

example for such a non-integrable function on $[0,1]$ [duplicate]

This is an exercise from the book "Real Analysis for Graduate Students": Find a non-negative function $f$ on $[0,1]$ such that $\lim_{t\rightarrow \infty} t m(\{x: f(x)\geq t\})=0$, but $f$ is not ...
2
votes
2answers
52 views

Properties of Lebesgue Integrals

If I have 2 Lebesgue Integrable functions $f,g$ defined on the same set A such that: $$ f > g \qquad \hbox{a.e on A}$$ Does this imply that: $$ \int_{A} f d\mu > \int_{A} g d\mu$$I'm not sure ...
2
votes
2answers
190 views

Showing that this integral diverges

I need to compute: $$ \lim_{n \to \infty} \int_{0}^{\infty} (1+x)^{np} \prod_{j=1}^{p} (1+xt_j)^{-n}dx $$ Where the the $t_j \in [0,1] \; \; \forall j$ How can I mathematically show that is is ...
3
votes
2answers
208 views

Finite function with infinite Lebesgue integral over any positive measure set

Is there a measurable function $f:\mathbb{R}\rightarrow [0,\infty)$ such that $$\int_A f\, \mathrm{d}\lambda=\infty$$ for any (measurable) set $A\subseteq\mathbb{R}$ with $\lambda(A)>0$. ...
1
vote
1answer
60 views

If $f \in L^1(-\infty, \infty)$ , and $G(u) = \frac{f(x+u) - f(x^+)}{\pi u}$, is $G(u) \in L^1(0^+, \infty)$ true?

To be more detailed, if function $f(x)$ satisfies $\int_{-\infty}^{\infty}|f(x)|dx < \infty$ and assume that $G(u) = \frac{f(x+u) - f(x^+)}{\pi u}$, is it true that $lim_{K \rightarrow \infty} ...
2
votes
1answer
25 views

Integral in $L^p$ spaces

Let $f: (0, \infty)\to \mathbb R$ be defined by $$f(x)=x^{-1/2}(1+|\ln x|)^{-1}.$$ Prove that $f\in L^2 (\mathbb R_{+}; m)\setminus L^p (\mathbb R_{+}; m)$ for all $p\in [1, \infty)\setminus \{2\}$. ...
0
votes
0answers
14 views

$\sum_{i=1}^{∞}\sum_{j=1}^{∞}a_{ij}=\sum_{j=1}^{∞}\sum_{i=1}^{∞}a_{ij}=\text{lim}_{r\to∞}\sum_{(i,j)\in rV}a_{ij}$

Let $a_{ij} ∈ \mathbb{C}$, $i, j = 1, 2, . . $ . If $\sum_{i=1}^{∞}\sum_{j=1}^{∞}|a_{ij}|<∞$ Then ...
1
vote
0answers
45 views

If a series of functions is integrable, then is it convergent for almost everywhere?

I get a question when I read Stein's "Real Analysis" on page 70. If a function is defined by a series of functions, and this series is integrable, then must the series (partial sums) be convergent ...
1
vote
1answer
33 views

proving integrability of a function

For a fixed $a\in \mathbb{R^n}$, we set $$\Gamma(a,x)=\frac{1}{2\pi}log|a-x|\ \ \ \ \text{for $n=2$}$$ $$\Gamma(a,x)=\frac{1}{\omega_n(2-n)}|a-x|^{2-n}\ \ \ \ \text{for $n\ge 3$}$$ where $\omega_n$ ...
1
vote
1answer
46 views

Density of $C_c^\infty$ in $W_0^{1,2}$

Let $\Omega \subset \mathbb{R}^N$ be a bounded open set and let $(f_n) \subset L^2(\Omega)$. Suppose there exists $f \in L^2(\Omega)$ such that $$\int_{\Omega} f_n \varphi \rightarrow \int_{\Omega} f ...
1
vote
1answer
28 views

Find the integral where , $f(x)$ is the decimal expansion of $x$.

Let , for each $x\in [0,1)$ $x=0.x_1x_2x_3...$ be the decimal expansion of $x$not eventually all $9's$. Define $f:[0,1)\to \mathbb R$ by $f(x)=x_1$ , the first digit in the expansion. Then ...
0
votes
0answers
34 views

Failure of Fubini's Theorem

Let $X=[1, \infty)$, $Y=[0, 1]$, $\mu=\nu=m$-the Lebesgue measure on $X$ and $Y$ and $\mathcal A=\mathcal B=\mathcal M$-the Lebesgue $\sigma$-algebra on $X$ and $Y$. Show that $f:X\times Y\to\mathbb ...
1
vote
1answer
29 views

Is the Lebesgue integral the same as the supremum of lower Darboux sums?

My textbook has a lot of definitions that look more or less the same thing to me so excuse my ignorance. It first defines a simple function as a function that can be written as ...
0
votes
0answers
33 views

Optimum point of $f(s) = \int_0^{\pi} \frac{ \exp(-s) y \cos(ky)}{s^2+y^2} \,dy $

Is it possible to find optimum point for the following function f(s) (i.e. $df/ds=0$): $$ f(s) = s e^{-s} \int_0^{a} \frac{ y \cos(\frac{\pi}{a} y)}{s^2+y^2} \,dy $$ or $$ f(s) = s e^{-s} ...
3
votes
0answers
62 views

Solution of $\int_0^{\pi} \frac{ y \cos y}{s^2+y^2} dy$

Is there a solution for the following integral (even in terms of Bessel or Struve functions)? $$ \int_0^{\pi} \frac{ y \cos(y)}{s^2+y^2} \,dy $$
0
votes
1answer
50 views

Behaviour of $L^1$ and $L^2$

In another proof I stumbled over the following question. Let $(e_n)$ be an ONB of $L^2$ on a compact domain and $f \in L^1.$(this is the point here, $f$ is a priori not in $L^2$).So we have $L^2 ...
1
vote
1answer
42 views

Integral of $au^2$ where $a$ is continuous and $u \in W_0^{1,2}(\Omega)$

Let $\Omega \subset \mathbb{R}^N$ be a smooth domain with bounded complement. Let $a \in C(\Omega)$ and let $u \in W_0^{1,2}(\Omega)$. Suppose that $a > 0$ in $\Omega$ and $\displaystyle ...
0
votes
0answers
34 views

What is the dual space of $L^2([0,1],[0,1])$

Suppose $L^2([0,1],[0,1])$ contains all Lebesgue square-integrable functions mapping from $[0,1]$ to $[0,1]$. Does the dual sapce be represesnted by all functions in $L^2([0,1],[0,1])$ or all ...
5
votes
0answers
50 views

$L^2$ convergence of this sequence

I am given the following sequence of functions $(f_m)_{m \in \mathbb{N}}$. They are defined by $$ f_m(x):=\left( \frac{e^{-ix}-1}{-ix} \right)^m \left( \sum_{l \in \mathbb{Z}} \frac{\left|e^{-ix}-1 ...
2
votes
0answers
53 views

Convergence of a subsequence of a subsequence of distribution functions

I'm trying to find a solution for the following problem: Let $(\mu_n)_{n\in\mathbb{N}}$ be a sequence of signed (Baire)-measures (of bounded variation) on $[a,b]$ and let $F_{\mu_n}(t):=\mu_n([a,t))$ ...
4
votes
1answer
52 views

Lebesgue integral of a strange function.

The problem statement is as follows: Let $f: [0, 2]\to \mathbb R_{+}$ be defined by $f(t)=m(\{x\in [0, \pi]: t\leq 1+\cos (3x)\leq 3t\}).$ Compute $\int_0^2 f(t)\,dt$. I'm not certain how to begin ...
1
vote
0answers
20 views

Convergence in distribution of distributions $p_n$ implies convergence in distribution of $s_n$?

Question Setup Suppose $p_n(x,y)$ is a sequence of probability densities on $\mathbb R^2$ and $q_n(x)$ is a sequence of densities on $\mathbb R$ such that \begin{align*} \int b(x,y) \ p_n(x,y) \ dx ...
0
votes
1answer
15 views

Obtaining Essential Range and Support of a Measurable Function from Estimate

The following is an old real analysis qual problem which I cannot solve. Problem. Let $f\geq 0$ be a measurable function on $\mathbb{R}^{n}$. Suppose there exists $C>0$ such that for all ...
4
votes
1answer
53 views

integrability of $f^3$ for some Lebesgue measurable function

I'm trying to solve the following problem from an old qualifying exam, but nothing I've tried has been successful, so any help would be greatly appreciated. Suppose $f: \mathbb{R} \rightarrow (0, ...
0
votes
1answer
22 views

Set of positive measures and Banach space

In measure theory i heard recently a statement in my class, which says that the set of all (positive) measures does not make a Banach space ( whereas the set of signed measures makes up a Banach space ...
2
votes
1answer
38 views

Integration by parts formula with Lebesgue Integral and distribution function

I'm struggling to find a solution for the following problem: Let $f$ be an absolutely continuos function on [a,b], let $\mu$ be a bounded Borel measure on [a,b], and let $\Phi_\mu(t)=\mu([a,t))$ with ...
1
vote
1answer
51 views

Evaluate limits:$\lim\limits_{n\to\infty}\int_0^nf_n^2\ dm$, $\lim\limits_{n\to\infty}\int_0^nf_n\ dm$ with $f_n(x)=\frac{e^{\sin(x^2/n)}}{1+x}$

So I am working through some practice problems, and on one of them I can't get the second part: For $x\in(0,\infty)$ and $n\in\{1,2,3,\dots\},$ let $f_n(x)=\frac{e^{\sin\left({x^2/n}\right)}}{1+x}.$ ...
1
vote
1answer
42 views

Problems in the integration limits to apply Fubini's theorem

If $f:(0,a)\rightarrow\mathbb{R}$ integrable function and $$g(x)=\int_{x}^a \dfrac{f(t)}{t}dt.$$ Then $g$ is integrable and $\int_{0}^a g(t)dt=\int_{0}^a f(t)dt$. I have to use Fubini's theorem but ...
2
votes
0answers
11 views

When can we move a Fréchet derivative under a Lebesgue integral?

Under what conditions can we move a Fréchet derivative under a Lebesgue integral? Specifically, when does $$ G'(x) = h\in X\mapsto \int_{\Omega} \left(F_x^\prime(x,t)h\right) \mu(dt) $$ where $$ ...
1
vote
1answer
25 views

L^p spaces are separable and complete but not compact?

Where is the mistake in my reasoning?: Let X be a separable metric space, then for every $p\in [1,\infty)$ and for every borel measure $\mu$ on $X$: $L^p_{\mu}(X)$ is separable. Therefore by a ...
4
votes
1answer
47 views

Example for the benefit from monotone convergence

I want to see a (preferably simple) example where I can apply monotone convergence to a sequence of functions $f_n$ but where I cant exchange limitation and integration in terms of the Riemann ...
-3
votes
1answer
30 views

how to construct a monotonic function on a closed interval which is discontinuous at each end points [closed]

How to construct a monotonic function on a [0,1] which is discontinuous at each end points?
3
votes
1answer
43 views

Intuition/proof that $E(X)= \int X(w) dP = \int x d\alpha$, where $\alpha$ is the cumulative distribution function of X

Looking for more intuition/help explaining the equivalence of the following two integrals. I know that the push-forward measure, or the CDF, of a random variable $X$ on a prob. space $(\Omega, \cal ...
1
vote
0answers
31 views

Can I approximate a measurable set with an open set for integration purposes?

I have a Lebesgue measurable function $f:X\rightarrow \mathbb{R}$ where $X\subset\mathbb{R}$. Is there an open set $X^O$ such that \begin{equation*} \int_X f=\int_{X^O} f \end{equation*} and $X^O$ is ...
1
vote
1answer
51 views

Intuition behind variance in terms of $L^P$ norms?

I've just started working through Varadhan's Probability lecture notes, and I was wondering if there's any intuitive connection between the variance formula and Holder's inequality/ $L^p$ norms in ...
3
votes
1answer
69 views

Proving that a trigonometric sum is in $L^2$

How can I use Parseval's identity to prove that $$f(x)=\sum_{k=1}^\infty \frac{\sin(kx)}{1+k}$$ is in $L^2(0,\pi)$? Thank you!
7
votes
2answers
425 views

Integrating over the naturals. What does it mean?

Let $F$ be the power set of $\Bbb{N}$ and consider the measurable space $(\Bbb{N}, F)$. Then what does it mean to take the integral with respect to the measure $\mu(A) = \sum_{a \in A} \frac{1}{a}$. ...
2
votes
0answers
13 views

$L_2((-2,2))$ function that has $L_1((-1,1))$ discrete derivative but not derivative

I am trying to find an example of a function $u\in L_2((-2,2))$ such that $||\delta_h(u)||_{L_1((-1,1))}$ is uniformly bounded in $0<|h|<1/2$ but $u'$ is not in $L_1((-1,1))$. Where ...
5
votes
2answers
54 views

dominated convergence for functions $\mathbb R^n\to\mathbb R^m$?

I do know the dominated convergence theorem for functions $f:\mathbb R^n\to\mathbb R$. Now let $U\subset\mathbb R^n$ and $f: U\to\mathbb R^m$. Is there any dominated convergence theorem for ...
2
votes
1answer
39 views

Find an example that the following equality doesn't apply

I need a sequence $(f_n)_{n\in\mathbb{N}}, f_n:X\to\mathbb{R}^\star$ which is measurable, $f_n \ngeq 0$ and doens't accept the following equality: $$\int_X\sum_{n=1}^\infty f_n \, d\mu = ...
0
votes
1answer
48 views

Hilbert space L2 - inner product

I have a problem with one exercise. I have to prove that $L^2$ space is Hilbertian. So I think that the best way is to check out inner product by definition of norm, so: \begin{equation*} ...
1
vote
1answer
45 views

If $\Omega\subseteq\mathbb{R}^n$ is bounded, then $\int_\Omega|x-y|^{1-n}\,d\lambda < \infty$

Let $\Omega\subseteq\mathbb{R}^n$ be bounded with $n\ge 2$ $\left|\;\cdot\;\right|$ be the euclidean norm $\lambda$ be the Lebesgue measure on the Borelian $\sigma$-algebra of $\mathbb{R}^n$ I ...