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Questions tagged [lebesgue-integral]

For questions about integration, where the theory is based on measures. It is almost always used together with the tag [measure-theory], and its aim is to specify questions about integrals, not only properties of the measure.

0
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2answers
26 views

Between proper integrals and improper integrals

I just started learning about improper integrals. Many of them are improper because the function evaluates to infinity at some point in their domains, e.g. $f(x)=1/x$ on the domain of $(0,1)$. My ...
1
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1answer
30 views

$L^p$ functions for $p$ in $[a,b]$

I have seen a theorem that assures that the set $A=\lbrace p\in [1,\infty]: u\in L^p(0,\infty)\rbrace$ is an interval. It is easy to find a function $u$ for which $A$ is the empty-set or $[1,\infty]$ ...
1
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0answers
19 views

Exponential decay and integration

I am confronted with the following problem: Let $\mu$ be a probability measure on $\mathbb{R}$. We wish to show that for any $p \in \mathbb{N}$ and $r \in \mathbb{R}$, the integral $$ F(r):= \...
0
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0answers
19 views

Proof a set has potitive measure

Let $S \subset \mathbb{R} \times \mathbb{R}$ such that $|S| > 0$. How to prove that $| \{ x-y, (x,y) \in S \} | >0$ ? (because I want to prove that $f(x)=0$ ae implies $f(x-y) = 0$ ae in (x,y)....
4
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1answer
18 views

Lebesgue measurable but not Riemann integrable

Every bounded function $f:[a,b]\to\mathbb R$, which is Riemann integrable, it is also Lebesgue integrable. On the other hand $$ g(x)=\left\{ \begin{array}{lll} 1 & \text{if} & x\in\mathbb Q\...
2
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1answer
31 views

Proving continuity and boundedness of the Fourier Transform

Let $f\in L^{1}(\mathbb R^d)$. Define the Fourier transform of $f$ by $$\hat{f}(y)=\int_{\mathbb R^d}f(x)e^{-2\pi ix\cdot y}\,dx\,\,\,\,(y\in\mathbb R^d).$$ Show that $\hat{f}:\mathbb R^d\to\mathbb{...
1
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0answers
20 views

Does $\int^{\infty}_{0}pa^{p-1}1_{\{|f|>a\}}(x)da=\int^{|f(x)|}_{0}pa^{p-1}da$?

Let $f\in L^p(\mathbb{R}^d)$ for $p\in [1,\infty)$. Show that $$\|f\|_p=\left(\int^{\infty}_{0}pa^{p-1}m\{|f|>a\} \right)^{1/p}$$ My attempt: We can use Fubini's theorem in the following way: \...
1
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0answers
30 views

The integral of a function is 0 in intervals, so the function is 0 everywhere

I am having trouble solving this problem of my course of measure and integration. Let $f$ an integrable function on the measure space $\mathbb{R}$, $L$, $\lambda$, where $L,\lambda$ is the Lebesgue-...
1
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0answers
27 views

Convolution of $L^{1}$ functions is well-defined

Let $f,g\in L^{1}(\mathbb R^d).$ The convolution of $f$ and $g$ is the function $f\ast g$ defined by $$(f\ast g)(x)=\int_{\mathbb R^d}f(x-y)g(y)\,dy.$$ Show that $(f\ast g)(x)$ is well-defined for a....
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0answers
12 views

Equality in distribution sense, implies equality almost everywhere

Let $\Omega \in \mathbb{R^n}$, (open set). and $f\in L^2(\Omega)$. If we had the equality in distribution sense : $u=f$ : $(1)$ Does it implies that $u\in L^2(\Omega)$ ? This is what I did : Let $...
0
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0answers
41 views

Simpler Proof for the Special Case

Suppose $1 \le p < \infty$ and $f_n ,f \in L^p[0,1]$. If $f_n \to f$ almost everywhere, then prove that $||f_n-f||_p \to 0$ iff $||f_n||_p \to ||f||_p$. The above problem does not come the real ...
0
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0answers
30 views

How to prove Dilation invariance of Lebesgue integral

Let $f\in L^{1}(\mathbb{R}^d), a_1,\dots,a_d>0$, and $a=(a_1,\dots,a_d)$. Define $$g(x)=f(a_1^{-1}x_1,\dots,a_d^{-1}x_d).$$ Show that $d\in L^{1}(\mathbb R^d)$ and that $$\int g=\left(\prod^{d}...
1
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1answer
51 views

If $a_n\to 0$, does $f(x+a_n)-f(x)\to 0$ in the $L^1$ norm?

I had this problem in my midterm exam the last week and still I can't handle it: It is true that if $f \in L^{1}(\mathbb{R})$, (that is, $f: \mathbb{R} \rightarrow \mathbb{R}$ is Lebesgue measurable, ...
5
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1answer
171 views

Counterexample for the monotone convergence theorem

Do you have a counterexample for the monotone convergence theorem when you omit the hypothesis that the sequence is increasing? I was thinking about the example where the sequence $f_n$ would approach ...
0
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2answers
27 views

Lebesgue integral over shrinking set is 0 as the set approaches measure 0 [on hold]

I had the following problem on my midterm, and am trying to figure out what the correct solution is when preparing for my final. Any suggestions are appreciated, thank you! Let $f$ be a fixed ...
1
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0answers
29 views

Integral over the set of leaders of a bounded measurable function

Let $f:[0,1]\rightarrow \mathbb{R}$ be a bounded measurable function. We say $t\in[0,1]$ is a leader if $\int_0^{\epsilon} f(t+s)\,\mathrm{d}s<0$ for some $\epsilon\in [0,1-t]$. Let $L\subset [0,1]$...
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0answers
51 views

Is the integral equal to zero?

Consider $$\int_{S_r} \left(\frac{\partial\phi}{\partial\eta}\right)^2 (x\cdot\eta)\ d\sigma $$ Where $S_r$ is the sphere of radius $r$ and centered at zero, $\phi\in L^2(\mathbb{R}^n)$ and $\eta$ is ...
3
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1answer
66 views

$\int_A fd\mu=0$ for all $A$ in a generator of the $\sigma$-algebra $\Rightarrow$ $f=0$ $\mu$-almost everywhere?

Assume I have a measurable signed function $f$ for which the integral with respect to a measure $\mu$, on some measurable sets $\mathcal{C}$ generating the sigma algebra $\mathcal{E}$, is zero (maybe ...
0
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2answers
65 views

Infinite integral that seems to diverge but question requires it does not.

The formula for half space solutions to the Laplace equation can be given: \begin{equation} u(x,y) = \int_{z\in \mathbb{R}} \frac{1}{\pi}\frac{y}{(x-z)^2+y^2}f(z) \end{equation} Use this formula to ...
0
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1answer
24 views

If f is Lebesgue integrable, then $g(x) = \int_x^{\infty} f(y) d \lambda (y)$ is continuous.

This is a true or false question, $f,g$ both $ \mathbb{R} \rightarrow \mathbb{R}$. I think it is right, but in the proof, I have to show for any positive $\epsilon$, there exists positive $\delta$, ...
0
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1answer
23 views

Why $\int^{\infty}_{0}1_{|f(x)|>\alpha}d\alpha=|f(x)|$?

$f$ is integrable on $\mathbb{R}^d$. For each $\alpha>0$, let $E_\alpha=\{x:|f(x)|>\alpha\}$. Show that $$\int_{\mathbb{R}^d}|f(x)|dx=\int^{\infty}_{0}m(E_\alpha)d\alpha$$ I saw a proof of ...
2
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0answers
22 views

How can I justify the change of Sum and Integral sign??

Let $f:\Bbb{D} \rightarrow G\subset\Bbb{C}$ where $f$ is analytic and also $f$ is an isomorphism between $\Bbb{D}$ and $G$. $f$ has the series expansion $f(z)=\sum_{n=0}^{\infty}c_n z^n$. I must show ...
0
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0answers
26 views

Lebesgue Integral of a Lipschitz function

Suppose $F$ is a closed set in $\mathbb{R}$, whose complement has finite measure, and let $\delta(x)$ denote the distance from a point $x$ to $F$. That is, $$ \delta(x) = \inf\{|x-y|: y\in{F}\} $$ ...
1
vote
1answer
28 views

Proving Tchebychev's Inequality

Suppose $f\geq 0$, and $f$ is lebesgue integrable. If $\alpha>0$ and $E_\alpha=\{x:f(x)>\alpha\}$, prove that $$ m(E_\alpha)\leq 1/\alpha \int f $$ My proof attempt Proof. Let the ...
3
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2answers
36 views

Measurability of maximal functions

I have been reading Fourier Analysis by J. Duoandikoetxea. I got stuck on proving the measurability of maximal functions. The general maximal function/operator in this book is from the following ...
1
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0answers
17 views

Proving $g_a\in L^1(\mathbb{R}^d) \iff a<-d$

Prove that $g_a \in L^1(\mathbb{R}^d) \iff a<-d.$ Where $$ g_a(x)=\begin{cases} |x|^a & \text{ if } |x|>1 \\ 0 & \text{ otherwise} \\ \end{cases}$$ We will be using the following ...
1
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0answers
33 views

If $1<p<\infty, p^{-1}+q^{-1}=1,f\in L^{1}(\mathbb{R}^d),$ then there are $g\in L^{p}(\mathbb{R}^d)$ and $h\in L^{q}(\mathbb{R}^d)$ such that $f=gh.$

If $1<p<\infty, p^{-1}+q^{-1}=1,$ and $f\in L^{1}(\mathbb{R}^d),$ show that there are $g\in L^{p}(\mathbb{R}^d)$ and $h\in L^{q}(\mathbb{R}^d)$ such that $f=gh.$ $\textbf{My Thoughts:}$ I ...
1
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0answers
17 views

Riemman-Stieltjes Integration-IVT

I need to show this, Suppose that $f,g: [a,b] \rightarrow \mathbb{R}$ are continous. Show that exist $\eta \in (a,b)$ such that $$g(\eta)\int_a^\eta f(x)dx=f(\eta)\int_\eta^b g(x)dx $$ I defined $...
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0answers
16 views

Fourier serie in L^1[0:2\pi] [closed]

Do you any funciton $f$ in $L^1[0;2\pi]$ different from $\sum_{k\in Z} c_{k}(f) exp(ikx)$ almost everywhere ? Thanks you !
2
votes
1answer
32 views

If $\|f_n\|_{p}\leq n^{-2}, f_n\in L^{p},\forall n\in\mathbb{N}$, does pointwise convergence of $f_n$ follow for a.e. $x\in\mathbb{R}$?

Let $p\in[1,\infty)$ and $(f_n)_{n\in\mathbb{N}}\subset L^{p}(\mathbb{R})$ such that $\|f_n\|_{p}\leq n^{-2}$ for all $n\in\mathbb{N}.$ Does $(f_n)_{n\in\mathbb{N}}$ necessarily converge pointwise a.e....
0
votes
1answer
32 views

Pull Limes out of integral

Let $f_\epsilon$ converge pointwise to $f$ for $\epsilon \rightarrow 0$ and let $f_\epsilon$ and $f$ be integrable. Can I swap integral and limes in this case? $$ \lim_{\epsilon \rightarrow 0} \int_\...
1
vote
1answer
20 views

Rectifiable curves in $\mathbb{R}^n$ have measure $0$, but line integral is nonzero.

I covered a bit of Lebesgue measure and came across the property that states that the integral of a function $f$ over some nullset $E$ is always $0$. If rectifiable curves have measure $0$ in $\mathbb{...
1
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1answer
30 views

If $f:\mathbb{R} \to [0,\infty)$ is Lebesgue measurable and $\int_{(n,n+1]}f dm = 0$ for all $n \in \mathbb{Z}$, then $\int_E fdm =0$ for all $E$.

Let $m$ denote the Lebesgue measure on the Lebesgue sigma algebra $\mathcal{M}$. If $f:\mathbb{R} \to [0,\infty)$ is Lebesgue measurable and $\int_{(n,n+1]}f dm = 0$ for all $n \in \mathbb{Z}$, then $\...
0
votes
1answer
24 views

If $d_f (t) : (0, \infty) \rightarrow [0,\infty]$ prove $d_f (t) \leq \bigg( \frac{||{f}||_{L^p(X)}}{t} \bigg)^p.$

Let $1 \leq p < \infty$ and suppose that $f \in L^{p}(X).$ Define the function $d_f (t) : (0, \infty) \rightarrow [0,\infty]$ by $$d_f(t)= \mu (\{ x \in X : |f(x)|>t\}).$$ For each $t>0,$ ...
1
vote
1answer
33 views

Prove $\int \bigg(\sum_{n=1}^{\infty} f_n \bigg) d \mu = \sum_{n=1}^{\infty} \bigg( \int f_n d \mu \bigg) $

Suppose that $\{ f_n \}_{n=1}^{\infty}$ is a sequence in $L^{+}(X, \mathcal{M}).$ Prove that $$\int \bigg(\sum_{n=1}^{\infty} f_n \bigg) d \mu = \sum_{n=1}^{\infty} \bigg( \int f_n d \mu \bigg) $$ I ...
6
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0answers
47 views

Gagliardo–Nirenberg–Sobolev inequality for weighted Sobolev space with exponential weights

Consider the weighted $L^p_\omega(\mathbb{R}^d)$ space on $\mathbb{R}^d$ be the set of Lebesgue measurable functions such that $$\|f\|_{L^p_\omega}=\int_{\mathbb{R}^d}|f|^p\omega_\mu(x)\,dx< \...
2
votes
1answer
29 views

Proving that $L^{p}(\mathbb{R}^d)\not\subset L^{\infty}(\mathbb{R}^d)$ for $1\leq p<\infty.$

Show that $L^{p}(\mathbb{R}^d)\not\subset L^{\infty}(\mathbb{R}^d)$ for all $p\in[1,\infty).$ My attempt at this problem: I define the following function for $x\in\mathbb{R}^d,$ $$f_a(x)=\begin{...
1
vote
1answer
64 views

notation - what does this $\wedge$ inside the integral mean?

I have this integral: $\int(\epsilon \wedge \|x\|^{2})\nu(dx)$ The $\wedge$ symbol means that I have to integrate $\|x\|^{2}$ when $\|x\|>\epsilon$ or when $\|x\|>1$?
1
vote
2answers
45 views

Prove that if $f\in L^{p}(E), 1\leq p<\infty, m(E)<\infty$, then $f\in L^{q}(E)$ for all $1\leq q\leq p$

Let $f\in L^{p}(\mathbb{R}^d),$ for $1\leq p<\infty,$ and $f$ is supported on a set $E$, of finite measure. Prove that $f\in L^{q}(\mathbb{R}^d)$ for all $1\leq q\leq p.$ Here are my thoughts so ...
1
vote
1answer
24 views

Prove that $\inf \left\{ \int_E f d\mu : \mu (E) \geq \alpha\right\} > 0$

The next problem appears in the book "Real analysis" by Thomson/Bruckner in the chapter of Integrable functions. My proof seems good to me but I'm not confident enough with my proofs in this subject. ...
0
votes
0answers
13 views

Square integrability of $ \exp \{-\int t(1-\hat{f}(s/(At)^{\beta})) \} dt$

I would like to prove that $$ \hat{f}_I(s) = \exp \left\{- \lambda_1 2 \pi \int_0^{\infty} t(1-\hat{f}_F (s/l(t))) dt \right\} $$ is square integrable provided that $\hat{f}_F$ is square integrable ...
-1
votes
1answer
30 views

$\lim\limits_{t\to\infty} \min\{t,f\} = f$, for $f\geq 0$.

Assume $f\in C^1_c(\mathbb{R}^n), f\geq 0$. Define $$ f_t:=\min\{t,f\},\quad \chi(t):=\left(\int_{\mathbb{R}^n} f_t^{\frac{n}{n-1}}dx\right)^{\frac{n-1}{n}}. $$ Then $\chi$ is nondecreasing on $(...
0
votes
1answer
25 views

Prove that if $f$ is measurable , $g$ is integrable and $|f| \leq |g| $ then $f$ is integrable

If $f$ is a measurable function, $g$ is an integrable function, and $|{f}| \leq |{g}|$ on $X$, prove that these imply $f$ is an integrable function and $$\int |{f}| d \mu \leq \int |{g}| d \mu.$$ $\...
2
votes
2answers
38 views

Constructing a probability space

I am trying to understand the proof that $$\mathbb Eg(X) = \int_\mathbb R g(x)d\mu_X(x),$$ where $X$ is a random variable on a probabiliby space $(\Omega, \mathcal F, \mathbb P)$. It starts with the ...
0
votes
1answer
20 views

Continuity of integration as a function of the set

How to prove that if function is integrable then the integral as a function of the set is continuous ? We assume that the function is nonnegative and function going from arbitrary measure space with ...
2
votes
1answer
58 views

Find a sequence $(f_n)_n$ convergent in $L^{p}(\mathbb{R}^d)$ s.t. $\lim\limits_{n\rightarrow\infty}f_n(x)$ doesn't exist for a.e. $x\in\mathbb{R}^d$.

Construct a sequence $(f_n)_n$ convergent in $L^{p}(\mathbb{R}^d)$ such that $\lim\limits_{n\rightarrow\infty}f_n(x)$ does not exist for a.e. $x\in\mathbb{R}^d$. My thought so far: I am trying to do ...
3
votes
1answer
47 views

If $f_n\rightarrow f$ in $L^p$-norm, $1\leq p<\infty$, then $\lim\limits_{n\rightarrow\infty}m(\{x\in\mathbb{R}^d:|f_n(x)-f(x)|>\varepsilon\})=0$

Let $1\leq p<\infty.$ Let $(f_n)_n$ be a sequence in $L^{p}(\mathbb{R}^d)$ that converges to some $f\in L^{p}(\mathbb{R}^d)$, in $L^p$-norm. Prove that for every $\varepsilon>0$ we have $$l=\...
0
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1answer
17 views

improper integrability the following function

Let $f:[1,\infty)\to\mathbb{R}$ be a function defined by $$f(x)=\left\{\begin{array}{rl} \frac{1}{x^{2}}, & x\in\mathbb{Q}\cap[1,\infty), \\ -\frac{1}{x^{2}}, & x\in\mathbb{Q}^{c}\cap[1,\infty)...
-1
votes
1answer
56 views

Lebesgue Measurable Functions

Let $(X,\mathcal{A},\mu)$ be a measure space with $\mu(X)=1$.$\;$ Suppose $\varphi:X \longrightarrow [0,1)$ is measurable. Prove that $$\lim_{a\to\infty}\int_{X}\varphi^{a}\:\mathrm{d}\mu=0.$$ My ...
1
vote
1answer
22 views

Two basic questions in Wasserstein spaces

We denote by $P (\mathbb{R}^{d})$ the space of probability measures on $\mathbb{R}^{d}$ and for $p\geqslant 1$ the Wasserstein space by \begin{equation*} P^p (\mathbb{R}^{d}) = \{ \mu \in P(\mathbb{R}...