0
votes
1answer
28 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 $f \in L^\infty(\Omega)$?
1
vote
1answer
32 views

One dimensional integrals in Green's theorem

I am trying to understand Green's theorem, but the problem is I don't know what is the definition of the integrals in the theorem. This is the expression that one proves to hold with some assumption ...
6
votes
3answers
111 views

Finding the integral of $\frac{x}{e^x + 1}$ [duplicate]

I've having some difficulty with finding this integral: $$ \int_0 ^{\infty} \frac{x}{e^x + 1}$$ Now usually I would use the monotone convergence theorem to write (using geometric series): $$f_n (x) ...
0
votes
2answers
75 views

Integral $=0$ implies function$=0$

Let $f:[a,b]\to\mathbb{R}$ be a mesurable function. How can we show that if $$\int_a ^xf(s)ds=0,$$ for all $x\in[a,b]$, then $f=0$.
3
votes
3answers
158 views

Are all measures Lebesgue-Stieltjes measures?

In our lecture we ran out of time, so our prof told us a few properties about measure: He said that a measure is $\sigma$-additive iff it has a right-side continuous function that it creates. And he ...
2
votes
1answer
46 views

Lebesgue differentiation theorem with two variables

We know that if $f:\mathbb{R}\to\mathbb{R}$ is continuous then $$\lim_{h\to 0}\frac{1}{h}\int_x^{x+h} f(s)ds=f(x).$$ But if we have $f:\mathbb{R}\times \mathbb{R}\to\mathbb{R}$, what kind of ...
1
vote
1answer
71 views

How to show that this function is differentiable?

Let $$\phi: \mathbb{R} \rightarrow \mathbb{\mathbb{C}}, s \mapsto \int_2^{\infty} \frac{e^{isx}}{x^2\ln(x)}dx$$, I want to show that this function is differentiable everywhere. Unfortunately, it ...
1
vote
1answer
49 views

The limit $\lim_{h\to0}\frac{1}{h}\int_0^hf(s) \, ds$

We know that if a function $f:\mathbb{R}\to\mathbb{R}$ is continuous then we have $$\lim_{h\to0}\frac{1}{h}\int_0^hf(s) \, ds=f(0).$$ What can we say if $f$ is continuous almost everywhere or ...
2
votes
1answer
37 views

Convex , then also Measurable

I was reading about Jensen's inequality and noticed that don't require $\phi$ to be measurable here: Wikipedia link. Therefore, I guess that being convex implies being measurable somehow, but I have ...
4
votes
1answer
112 views

Show $\lim\limits_{a \rightarrow + \infty} \int_0^{\infty} \frac{1}{1+y^2}e^{-ay} dy =0 $

Need to prove $\lim\limits_{a \rightarrow + \infty} \int_0^{\infty} \frac{1}{1+y^2}e^{-ay} dy =0 $ and $\lim\limits_{a \rightarrow + \infty} \int_0^{\infty} \frac{y}{1+y^2}e^{-ay} dy =0 $ Can ...
1
vote
1answer
35 views

Is $l^p$ closed in $L^p$?

Let's assume we have a subspace $X$ of $L^p$ and we know that $X \cong l^p$(this should just mean isomorph no isometry is assumed here). Can we infer from this that $X$ is closed? I have just read a ...
2
votes
2answers
131 views

Biorthogonal functions in $L^p$

I asked one question that is already answered: 1.) I have a question about Lemma 9.5 on page 93/94 reference. It's about the part of the proof where the sequence of $(g_n^*)$ are introduced. I don't ...
1
vote
0answers
48 views

What means: is equivalent to?

I found the following theorem: Let $(f_n)$ be a sequence of norm one functions in $L^p, p \in [1, \infty)$. If $\lambda(supp(f_n)) \rightarrow 0$, then some subsequence of $(f_n)$ is equivalent to a ...
1
vote
1answer
31 views

Weak absolute continuity of measures

I want to show that if we have a function $f \in L^p$ sucht that $||f||_p =1$. Define a new measure $\mu$ by $$\mu(A):=\int_A |f(x)|^p dm(x).$$ Then $\forall \epsilon > 0 \ \ \exists ...
0
votes
1answer
39 views

Integrability of sums of Dirac deltas

this is my first post in the forum and I am an engineer, so I apologize in advance if my question is not clearly stated. Consider the function $f(x)=\sum_{i=1}^N a_i\delta(x-x_i)$ where ...
12
votes
2answers
225 views

Topology of convergence in measure

Currently I am doing some measure theory (on $X=[0,1]$ with the Borel-Sigma algebra and the Lebesgue measure), and I am looking at sets $A \subset L^p$, such that for all $q \in (0,p)$, the topologies ...
1
vote
1answer
122 views

Continuity of a parametric integral (where the integrated function is discontinuous)

For all $t\in\mathbb{R}$ consider $$F(t):=\int_\mathbb{R}e^{-x^2/2}\log|t+e^x|\,dx \;.$$ I managed to show that $F(t)$ is well-defined and finite for every $t$. I would like to show that $F$ is ...
2
votes
1answer
60 views

Fundamental Theorem of Calculus for Riemann and Lebesgue

Quick question regarding the second part of the Fundamental Theroem of Calculus in terms of Riemann and Lebesgue Integration: In terms of applying the second part of fundamental theorem of calculus, ...
0
votes
1answer
62 views

$f '$ is not Lebesgue integrable on $[-1,1]$

Let f be that function from R to R defined by f(x)= 0 if x=0 x^2 sin(1/x) if x not = 0 show that the function f' is ...
1
vote
1answer
92 views

Understanding Lebesgue Integration

I have started studying Lebesgue integration and I have a few of questions regarding the Lebesgue integral: In the wikipedia entry of "Lebesgue integration" they define the Lebesgue integral as: ...
1
vote
1answer
46 views

For which $\alpha \in \mathbb{R}$ does $\int_{\mathbb{R}^n} \big(1+|x|\big)^{\!-\alpha} \mathrm{d}x$ exist?

I assume only $\alpha \gt 1$ gives $\int_{\mathbb{R}^n} (1+|x|)^{-\alpha} \mathrm{d}x \lt \infty$ (simply because this is true for $n=1$). I also assume some clever transformation could be used for ...
3
votes
1answer
74 views

Why is $g_n:=\inf\{f_n,f_{n+1},…\}$ integrable if $f_n$ are?

This question is motivated by the proof of the Fatou's lemma. My text defines $g_n:=\inf\{f_n,f_{n+1},\ldots\}$ and states that it's Lebesgue integrable (each $f_n$ is). We proved that point-wise ...
5
votes
3answers
535 views

Prove that function is not Lebesgue integrable

Prove that function $f(x,y)=\dfrac{1}{x^2+y^2}$ is not Lebesgue integrable on $A=(0,1]\times(0,1]$. To my knowledge the fastest way to do it is to use Fubini's theorem. From what I would get: ...
1
vote
1answer
77 views

Hölder's inequality. Understanding proof?!

I know how most standard textbooks show that $||f*g||_r \le ||f||_p||g||_q$ with $\frac{1}{r}+1=\frac{1}{p}+ \frac{1}{q}$, but I found a book where the hint $|f(x-y)g(y)|\le ...
7
votes
0answers
265 views

Egorov's theorem for this Lebesgue integral

I want to prove Egorov's theorem using this Lebesgue integral defined by the upper integral $$\int^*f:=\left\{\int h ; h \ge f \text{ and h upper-continuous }\right\}$$ $$\int_*f:=\left\{\int h ; h ...
3
votes
2answers
74 views

Shift Operator not continuous

Let $X=L^1(\mathbb{R}^n)$ and $T:\mathbb{R} \rightarrow L(X)$, such that $(T(\tau)f)(t):=f(t+\tau)$. The question is: Is $T$ continuous? Well, my idea was the following: $|\tau_1-\tau_2| \le \delta ...
1
vote
2answers
70 views

Finding a limit of an integral

I am trying to find the following limit. Let $X = [0,\infty)$ and $\mathbb B$ denote the Borel subsets in $[0,\infty)$, $\lambda$ the Lebesgue measure. Let $f_n : [0, \infty) \to \mathbb R$ be given ...
1
vote
1answer
165 views

Understanding Stokes' theorem

Stokes' theorem( here I am only talking about the special $\mathbb{R}^3$ case) contains a line integral $\int_{\partial S} \langle f, \tau \rangle ds$. (Actually, I would be confident if somebody ...
3
votes
2answers
103 views

Intuition for Lebesgue integration

I have started doing Lebesgue integration and I just want to clarify one thing to start with. Very loosely speaking, with Riemann integration we partition the domain into $n$ intervals, and then we ...
8
votes
1answer
113 views

A dominated convergence theorem applied to $e$ number definition

I want to show that: $$\lim_{n\to\infty}\left(1+\frac{1}{n}\right)^n=\sum_{k=0}^\infty \frac{1}{k!}.$$ By the binomial theorem $$\lim_{n\to\infty}\left(1+\frac{1}{n}\right)^n = ...
1
vote
1answer
175 views

Lebesgue outer measure in dilation invariant?

For $A \subseteq \mathbb{R}$. We know that $m^*(A + t) = m^*(A)$. but Does it follow that $$ m^*(\lambda A) = \lambda m^*(A)$$??? for $\lambda > 0$. Here $m*: P( \mathbb{R}) \to [0, \infty ] $ is ...
2
votes
1answer
83 views

integral over almost sure existing derivatives

Let $f$ and $g$ and $f-g$ be real-valued, Lipschitz functions, with Lipschitz constant smaller or equal to 1, on $[a,b]$ whose derivatives are positive and exist only $\lambda$-almost surely. Does ...
5
votes
2answers
1k views

Is Dirichlet function Riemann integrable?

"Dirichlet function" is meant to be the characteristic function of rational numbers on $[a,b]\subset\mathbb{R}$. On one hand, a function on $[a,b]$ is Riemann integrable if and only if it is bounded ...
3
votes
1answer
107 views

Integral inequality using positive and negative parts

Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a measurable (with respect to the Lebesgue measure), $\pi$-periodic function which is Lebesgue integrable over $[0,\pi]$. Moreover assume that ...
1
vote
0answers
30 views

The tightest bound on an integral

Consider a polynomial $p(x)$ where $p(x)>0$ for $x\in(0,1)$ and $p(0)=0$. Let $s(x)$ be an increasing analytic function such that $s(0)=0$ and $s(1)=1$. I am interested to bound the following ...
0
votes
1answer
143 views

Prove convergence of improper integral using change of variable.

This may be trivial, but I could use some help... Consider a real function $f: (0,1) \rightarrow \mathbb{R}$, continuous, positive, but not necessarily bounded. Let $g: [0,1] \rightarrow [0,1]$ be a ...
1
vote
1answer
59 views

Lebesgue integration— why are contours >t and not =t?

I read the Wiki on Lebesgue integration (http://en.wikipedia.org/wiki/Lebesgue_integration) and it says that the integral can be rewritten as the sum of volumes of equal-height contours: The volume ...
1
vote
3answers
151 views

How should I calculate the Lebesgue integral of logarithm function from zero to infinity?

Does the area under the $\ln(x)$ in $(0,+\infty)$ is measurable? If yes, how can I calculate it?
5
votes
3answers
188 views

$ \int^{\infty}_0 |\frac{1}{(1+x)\sqrt x}|^p ~ \mathrm dx < \infty \implies p=?$

If $ f(x) = \frac{1}{(1+x)\sqrt x} $ how to find all $ p > 0 $ such that $$ \int^{\infty}_0 |f(x)|^p dx < \infty $$ The integral is with respect to lebesgue measure. Any solution or hints would ...
2
votes
1answer
57 views

The Lebesgue Theory basic Application , get stuck

Ok, I am working on a very easy question but I get stuck when I trying to justify my answer. I know that, in order to use Lebesgue's dominated Convergence Theorem, there are two conditions that we ...
1
vote
2answers
333 views

Integrability of Derivative of a Continuous Function

Let $f$ be continuous on $[a,b]$ and has finite derivative a.e. on $[a,b]$. Let $f_n(x)=n[f(x+1/n)-f(x)] $ s.t. $f_n$ be uniformly integrable on $[a,b]$. I want to show : $f'$ is Lebesgue integrable. ...
2
votes
1answer
112 views

using sup of an unbounded function

Is what I'm doing valid if we don't have any information on boundedness of $f$ or $f_n$? let $X$ be a finite measure space and $\{f_n\}$ be a sequence of nonnegative integrable functions, $f_n ...
3
votes
0answers
272 views

Lebesgue Integration fundamental questions

My question involves the definition of the Lebesgue integral. Most colloquial definitions I've read follow (2), in that f*(t) is the "length" of one of the horizontal rectangles and dt is the ...
1
vote
0answers
350 views

Extended Riemann integrability of a non-negative function implies Lebesgue integrability?

Let $f$ be a bounded function on a finite interval $[a, b]$ of the real line. If $f$ is Riemann integrable, we denote its Riemann integral by $\mathcal{R}(f , [a, b])$. It is well known that $f$ is ...