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

1
vote
1answer
20 views

Monotone Convergence theorem for monotone decreasing sequences

Short question: (Just an example. I want to know if similar thoughts can be used for other sequences of functions) If I want to evaluate $\lim_{n\rightarrow \infty}\int_{[0,1]}-nxdx$, I can't do that ...
0
votes
1answer
53 views

Complex Measures: Polynomials

Given the complex plane $\mathbb{C}$. Consider a complex measure: $$\mu:\mathcal{B}(\mathbb{C})\to\mathbb{C}:\quad\operatorname{supp}\mu\subseteq\overline{B_r}$$ Then one has: ...
0
votes
1answer
46 views

Interpolation between derivatives

I am trying to prove the following: Let $u \in W^{2,2}(\mathbb R)$. Then $\| u^\prime \|_{L^2}^2 \leq \| u \|_{L^2} \| u^{\prime \prime} \|_{L^2}$ holds (these are meant to be weak derivatives). ...
4
votes
1answer
58 views

What is so good about the $L^2$-norm of the second derivative being small?

One of the main properties of cubic splines is the minimality property which basically means that if $s$ (cubic spline) and $g$ (some other function) interpolate $f$ in a certain way then $$\Vert s'' ...
0
votes
0answers
31 views

Relation between $\mathcal{L}_{1}$ and $\mathcal{L}_{2}$ Norms of a Lebesgue function

Can $\mathcal{L}_{1}$ norm of a function $f(t)$ be related with its $\mathcal{L}_{2}$ norm as $||f||_{1} \leq ||f||_{\infty} ||f||_{2}$ or something like this?
-1
votes
0answers
25 views

On the proof of Generalized Holder Inequality:

Can it be proved using generalised Holder inequality (i.e., $\frac{1}{p} + \frac{1}{q} \leq 1$) that $|<fg>|_{1} \leq |f|_{\infty} ||g||_{2}$ where $fg \in \mathcal{L}_{1}, \; f \in ...
2
votes
1answer
30 views

monotonic linear functional on $C_+(X)$

Let $X$ be a compact metric space. Let $C_{+}(X)$ be the set of all continuous non negative functions on $X$. Let $\lambda : C_{+}(X) \to [0,\infty)$ such that ...
0
votes
1answer
23 views

Does this contravene the dominated convergence theorem?

The function $f_n$: $\mathbb{R}\rightarrow \mathbb{R}$ is defined by $f_n(x):=\chi_{[0,\infty)}(x)\frac{1}{n}\exp(-\frac{x}{n})$, where $n\in \mathbb{N}$. $f(x):=\lim\limits_{n \rightarrow ...
0
votes
1answer
30 views

How to show $\int_X \sum_{n=1}^\infty f_n \, d\mu = \sum_{n=1}^\infty \int_X f_n \, d\mu$

$\{f_n\}$ are nonnegative monotonic increasing functions. Show that $$ \int_X \sum_{n=1}^\infty f_n \, d\mu = \sum_{n=1}^\infty \int_X f_n \, d\mu $$ Can someone give me a hint on how to show this? ...
3
votes
1answer
77 views

A Riemann Integrability Question

Define $f:\mathbb{R} \rightarrow \mathbb{R}$. For any fixed closed interval $[a,b] $,$f(x) $ is $Riemann$ integrable on $[a,b].$ Show that $\forall ...
1
vote
0answers
21 views

How can I show that the function in my problem is lebesgue integrable?

The function $f_n$: $\mathbb{R}\rightarrow \mathbb{R}$ is defined by $f_n(x):=\chi_{[0,\infty)}(x)\frac{1}{n}\exp(-\frac{x}{n})$, where $n\in \mathbb{N}$. Now: How can I show that ...
2
votes
1answer
33 views

Regularity of $\phi$ in order that $\int g_h \phi \,dx \to \phi(0)$

Define the sequence of functions $(g_h)_h$ where $$g_h(x):= h\, \chi_{[0,1/h]}(x)$$ and the sequence of measures $$(\mu_h(dx))_h:= g_h(x)\,dx.$$ We want to show that $\mu_h ...
0
votes
1answer
29 views

weak topology and weak* topology on $L^1, L^{\infty}$

Suppose $L^1(I)$ is the primal space and $L^{\infty}(I)$ is the dual. Could I simultaneously define weak topology on $L^1(I)$ with respect to $L^{\infty}(I)$ and define weak or weak* topology on ...
0
votes
1answer
33 views

Prove that for almost every $x\in$ $\mathbb{R}$ , $\lim_{n\to\infty}n^{-p}f(nx)=0$ .

Let $f\in$$L^1(\mathbb{R})$ and let $p>0$ . Prove that for almost every $x\in$$\mathbb{R}$ , $\lim_{n\to\infty}n^{-p}f(nx)=0$.
0
votes
0answers
31 views

Completion of C(I) to $L^{2}(I)$ for some arbitrary interval I

As $L^{2}(I)$ is the completion of C(I), without too many issues (as $L^{2}(I)$ is a space of equivalence classes with equivalence relation defined as functions equivalent if differ at only finitely ...
3
votes
1answer
31 views

Monotone convergence theorem in a special case

Suppose $C^{+}[0,1]$ be the set of all continuous functions with domain $[0,1]$ taking non-negative values only. Let $\lambda : C^{+}[0,1] \to [0,\infty)$ be a map that satisfies ...
2
votes
1answer
25 views

Equivalences of weak convergence in $\mathcal{L}_p$ spaces with the Lebesgue measure

Let $\Omega =(0,1)$, and $f,f_n\in \mathcal{L}_p(\lambda)$. Prove that if $\sup_n{\| f_n \|}<\infty$ and $$\int_{(0,t]}f_n \, \,\mathrm{d}\lambda\rightarrow \int_{(0,t]}f \, ...
2
votes
1answer
36 views

Benefit from measure theory

With your help I want to list the benefits from measure theory and the lebesgue integral. (Advantages to the Riemann integral) What I know: With the Lebesgue integral we need less requirements to ...
0
votes
1answer
33 views

The dual space of a product space

Suppose $\prod^n L^1(I) $ is the product space of n $L^1$ integrable functions. What would be its dual space of continous linear functionals? would it be $\prod^n L^{\infty}(I)$? Do I need the norm ...
3
votes
1answer
63 views

Simple question about the supremum of lebesgue integrals

Do we have the following equality? Let $h$ be measurable and non-negative, $f$ another measurable function and $g$ a step-function, then: $$\sup_g\left\{\int_X(fg)\,d\mu:0\leq g\leq ...
4
votes
3answers
78 views

If $f$ is in $L^{p}$, prove that $\lim \int_{x}^{x+1} f(t) dt = 0$

If $f$ is in $L^p$, prove that $\lim_{x \to \infty} \int_{x}^{x+1} f(t) dt = 0$ It is easy to think that integration must be vanish as $x \to \infty $ but I cannot write them with math. Suppose ...
1
vote
0answers
24 views

Video lectures on Measure and Integration

Does anyone know a good online lecture series on measure theory and Lebesgue integration? I looked at the MIT open courseware but I could find only lecture notes. I am interested in lectures on this ...
0
votes
2answers
28 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
32 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
21 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
55 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
57 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
37 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
45 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
48 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
190 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
41 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
40 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
27 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
30 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
25 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
60 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
40 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
33 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
49 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
52 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 $F_{\mu_n}(t):=\mu_n([a,t))$ ...