0
votes
2answers
40 views

$L^{\infty} (X, \mu)$ is not separable? [on hold]

Show that $L^{\infty} (X,\mu)$ is not separable if $X$ contains sequences of disjoint sets of strictly positive measure?
2
votes
2answers
55 views

Is a $L^1$-function which is linear near the origin in $L^p$?

Suppose you have a function $f$ on $\mathbb{R}$, such that $$\int_{-\infty}^{\infty} | f(x) | \, \mathrm{d} x < \infty$$ and $$\int_{-u}^u |f(x)| \, \mathrm{d} x = \mathcal{O}(u)$$ for $u \to 0$. ...
1
vote
0answers
47 views

When does an integral operator belong to the Schatten - von Neumann class in terms of its kernel?

It is well known that an integral operator $X: L^2(\mu)\to L^2(\nu)$ with kernel $k(x, y)$ belongs to the Schatten -- von Neumann class $\mathfrak S_2$ if and only if $\int |k(x, y)|^2\, d\mu(x)\, ...
4
votes
1answer
81 views

Proof that $(L^1)\neq(L^\infty)^\ast$

I have seen a "proof" that $L^1\neq(L^\infty)^\ast$ which goes as follows: show that there is an element of $(L^\infty)^\ast$ which is not in the image of the canonical map ...
3
votes
1answer
29 views

The existence of conditional expectation with respect to a sub-$\sigma$-algebra

I was trying to solve the exercise 3.17 from the book of real analysis by Folland and I've found a problem. The first part of the exercise is the following: Let $(X, M, \mu) $ be a $\sigma$-finite ...
3
votes
0answers
32 views

Can we expect, $h\ast \mu \in L^{2}(\mathbb R, (1+|x|^{2})^{s})$ for $h\in \mathcal{S}(\mathbb R), \mu\in M(\mathbb R)$ and $s>1/2$?

We put, $M(\mathbb R)=$ The space of complex bounded Borel measure on $\mathbb R$ [With each complex Borel measure $\mu$ on $\mathbb R$ there is associated a set function $|\mu|,$ the total variation ...
0
votes
2answers
72 views

Uniform integrability of a function in $L^1$

A collection of functions $(\phi_i)_{i\in I}\in L^1(\mu)$ is called uniformly integrable if given $\epsilon>0$ there exists $\delta>0$ such that : $$\int_E|\phi_i|d\mu<\epsilon~~~~\forall ...
3
votes
2answers
45 views

Convergence in $L^p$ by using Holder's inequality

Let $1\lt p \lt \infty$ and $f\in L_p[0,\infty )$. Show that a) $$\left\vert\int_0^x f(t)\,dt\right\vert\le\|f\|_px^{1-\frac{1}{p}},$$ for $x\gt 0$. b) $$\lim_{x\to \infty} ...
2
votes
1answer
43 views

Need help with application of Hardy-Littlewood inequality (Marcinkiewicz space and distribution functions)

I am going over this work here. I couldn't understand the equality where the Hardy-Littlewood inequality is used. I think $\delta$ here is a weight so we can take it to be $1$ for simplicity. Would ...
5
votes
1answer
58 views

$xf''(x) , xf', f \in L^{2}$ is $f' \in L^{1}$?

I am stuck on the following problem. I have a function $f$ such that $f$ is bounded on $(0,1)$, $xf'(x)$ is bounded on $(0,1)$, $f \in L^{2}(0,1)$, $xf' \in L^{2}(0,1)$, and $xf'' \in L^{2}(0,1)$. ...
1
vote
1answer
26 views

Don't understand a $L^\infty$ bound argument involving measure of set

I'm trying to understand the proof of Proposition 2.2, part 2 of this paper. this is where I am stuck. For any $k > 0$, we have $$k^{\frac{2(N+1)}{N}}|\{|u|^m > k\}| \leq ...
3
votes
2answers
37 views

For $1 \leq r < p < \infty$ prove the continuous injection of $L^p([0, 1])$ into $L^r([0, 1])$.

For $1 \leq r < p < \infty$ prove the continuous injection of $L^p([0, 1])$ into $L^r([0, 1])$. I am having a hard time starting. Any suggestions. I tried a straight forward approach. That ...
3
votes
1answer
68 views

$f$ is in $L^p$ iff sum is finite

Let $p\in [1,\infty)$.Prove that $f\in L^p(\mu)$ if and only if $\sum_{n=1}^\infty(2^n)^p\mu (\{x:|f(x)|\gt2^n\})\lt \infty.$ My idea, I assume measure is finite, I wrote ...
3
votes
1answer
19 views

$L^p$ integral on every measurable subset of $\Bbb R$

Suppose $f:\Bbb R \to \Bbb R$ is in $L^p$ for some $p>1$ and also in $L^1$. Prove there exist constants $c>0$ an $\alpha \in (0,1)$ such that $\int_A|f(x)|dx\le cm(A)^{\alpha}$, for every ...
1
vote
1answer
23 views

On finite measurable space $X$, the whole of $L^p(X)$ is closed in $L^1(X)$ iff there is a const $C$ st $||f||_p\leq C||f||_1, \forall f \in L^p(X)$

On finite measurable space $(X, \mathcal{M}, \mu)$, the whole of $L^p(X, \mu)(p>1)$ is closed in $L^1(X,\mu)$ iff there is a const $C$ st $||f||_p\leq C||f||_1, \forall f\in L^p(X)$, iff both ...
1
vote
0answers
52 views

If $f_{n}\rightharpoonup \bar{f}$ and $f_{n}(x) \rightarrow f(x)$ pointwise a.e., then is $\bar{f} = f$ a.e.? [duplicate]

Suppose $f_{n}$ is a sequence of functions in $L^{p}(\mathbb{R}^{d})$ such that $\|f_{n}\|_{L^{p}} \leq 1$ for all $n$ and $f_{n}(x) \rightarrow f(x)$ pointwise almost everywhere as $n \rightarrow ...
0
votes
1answer
38 views

When does the convergence of the regularization of a function is decreasing?

Hi everyone: Let $\theta(x)$ equal $k\exp\left(-\frac{1}{1-\|x\|^2} \right)$ if $\|x\|<1$, and equal $0$ if $\|x\|\geq1.$ Here $\|\cdot\|$ designates the Euclidean norm in $\mathbb{R}^n$, and the ...
4
votes
2answers
46 views

p-norm of a function

Let $f\in L^1(\mu)\cap L^\infty(\mu)$. I have proved for any $1<p<\infty$, $f\in L^p(\mu)$, $w(p)=||f||_p$ is continuous w.r.t. $p$, and $\lim_{p\to \infty}||f||_p=||f||_\infty$. Is $w(p)$ ...
2
votes
1answer
69 views

An amazing inequality of the integration of two functions.

Let $f:[0,1]\longrightarrow\mathbb{R}$ be measurable and $g\in L^1[0,1]$ such that for all $t>0$, $$ \int_{|f(x)|>t}|g(x)|~\mathrm{d}x\leq \frac{3}{t^2}. $$ Prove that for $1<p<2$, $$ ...
0
votes
1answer
44 views

Prove an identity without using Hölder's inequality

How to prove the following without using Hölder's inequality : $$ \|f\|_{p} = \sup_{\|g\|_q =1} \int |fg| d\mu ; \frac{1}{p} + \frac{1}{q} =1$$
3
votes
0answers
54 views

Is $L^p$ separable?

Whether a $L^p(X,\mu)$ space is separable? I understand that the answer depends on $p$ and $X$. It seems to me that it is separable when $1\leq p < \infty, X=\mathbb{R}^n$ or $X=\mathbb{N}$. ...
2
votes
2answers
59 views

Properties of a sequence of iid rv's

I cannot do part a), and Im fairly sure that b) and c) will follow from it. If possible could I please have a solution to part a) and hints if you feel necessary to parts b) and c).
2
votes
1answer
46 views

Increasing sequence of closed subspaces of $L^2$

Not sure how to start on this question. Writing out a decomposition didnt seem to lead to anything. I think you might have to guess the answer the question first before starting a proof. Please ...
2
votes
1answer
26 views

$L_p$ spaces and tail estimates

I can prove the main identity in this question. Not sure how the "and deduce" bit works. I think $O(\lambda^{-q})$ is some kind of tail estimate.
1
vote
0answers
52 views

Hardy Littlewood maximal function and integral comparison.

Define the Hardy Littlewood maximal function $$g^*(y)=\sup \left\{\frac{1}{|B|}\int_B|g(x)|dx:B\text{ is any open ball containing y}\right\}.$$ For given $x_i,r_i,a_i$, first I have shown that ...
1
vote
1answer
38 views

$L^p$ norm of a measurable function is bounded by its operation on step functions

Let $1\leq p<\infty$, $1/p+1/q=1$. Let $f$ be a measurable function on $[0,1]$ such that for all step functions $g$ on $[0,1]$ $$ \left|\int_0^1 fg d\mu\right|\leq \|g\|_q. $$ Prove that ...
2
votes
1answer
46 views

a condition given by step functions implies the condition holds for L^q space

Let $1\leq p<\infty$, $1/p+1/q=1$. Let $f$ be a measurable function on $[0,1]$ such that for all step functions $g$ on $[0,1]$, $$ |\int_0^1 fg d\mu|\leq ||g||_q. $$ Prove $||f||_p\leq 1$. How ...
0
votes
0answers
33 views

$L^{p}$ convergence [duplicate]

So... i am a little rusty with measure theory and i need some help with this exercise, if anyone can give me an idea to start Let $(X,\mu)$ a measure space. Let $f_n:X\rightarrow \mathbb{C}$ a ...
0
votes
2answers
23 views

Exercise on abstract integration

Let $f_n$ be a sequence of nonnegative functions defined on $\mathbb{R}^N$ such that $f_n \rightarrow f $ almost everywhere on $\mathbb{R}^N$ and such that $$\int_{\mathbb{R}^N} f_n \rightarrow ...
4
votes
1answer
107 views

Is $ L^{\infty} $ a direct limit or inverse limit of the directed system $ (L^p , i_{p}^q )_{p,q \in [1 , + \infty [ } $?

Let $X$ be a finite measure space. Then, for any $ 1≤p<q≤+∞ $ : $ L^q(X,B,m)⊂L^p(X,B,m) $. I would like to know if the space $ L^{\infty} ( X , B , m ) $ is the direct limit or the inverse limit of ...
1
vote
1answer
38 views

Dual of $L^1$ when measure is the counting measure [closed]

Let $X$ be an uncountable set, $\mu$ the counting measure on $X$ and $\mathcal{M}$ the $\sigma$- algebra of countable or co-countable sets. How can I prove that the dual of $L^1(\mu|\mathcal{M})$ is ...
1
vote
1answer
49 views

Prove that a relatively compact subset of $L^p$ is bounded.

Let $p\in [1,\infty)$, $A\subset L^p(\mathbb R^m)$ relatively compact and $\lambda^m$ be the Lebesgue measure on $\mathbb R^m$. Prove: a) $A$ is bounded. b) $\lim_{y \to 0}\sup_{f \in A} ...
1
vote
0answers
39 views

Convergence of product of continuous functions and test functions

I suspect the following result is true but I"m not sure how to go about proving: It is given that $\Omega \subset \mathbb{R}^{n}$ is an open bounded, connected domain.(Not sure if theses conditions ...
3
votes
1answer
68 views

can $L^p$ norm convergence and pointwise monotonic imply pointwise convergence?

Let $(f_n)_{n=1}^\infty$ be a sequence of measurable function such that $\lim_{n\to\infty}||f_n-f||_p=0$. If for any $x\in \Omega$, $\{f_{n}(x)\}_{n=1}^\infty$ is a monotonic sequence, can we deduce ...
3
votes
1answer
34 views

Application of weak $L^p$ estimate besides for proving boundedness of some linear operator

For all $1\leq p< \infty$, weak-$L^p(\mathbb{R}^d)$ space is defined as a set of all functions $f$ such that $$\gamma^p|\{x\in \mathbb{R}^d: |f(x)|>\gamma\}|<\infty$$ for every ...
1
vote
1answer
53 views

Is Lp space complete with this norm?

Let $E$ be a measurable set of finite measure and $1\leq a<b<\infty$. Consider the $L^b(E)$ space normed by $L^a$ norm. Is this space a Banach space? I think this is wrong, so I tried to find a ...
2
votes
1answer
41 views

function $L_p$ iff $1\leq p<2$

Let $X=<0,1>$, take the borel sigma algebra, and the lebesgue measure. Consider $g(x)=\dfrac{1}{x^{\frac{1}{2}}}$. Show that $g\in L_p$ iff $1\leq p<2$. I have done this: ...
2
votes
0answers
68 views

Does absolute continuity of measures imply a relation between the $L_p$ spaces?

Say $(X,\mathcal{B},\mu)$ is some measure space, and let $\sigma$ be some other measure on $(X,\mathcal{B})$ such that $\sigma\ll\mu$. What can one say about the relation between $L_p(\mu)$ and ...
1
vote
0answers
33 views

Convergence in $L^1$ and $L^p$ [duplicate]

Assume: (a) $\{f_n\} \subset L^p$, (b) $f_n \to f \text{ }\mu\text{-a.e.}$ and (c) $\|f_n\|_{L^p} \to \|f\|_{L^p}\to0$. Then show that $$\|f_n - f \|_{L^p} \to 0$$ using Fatou's Lemma, first for ...
1
vote
0answers
47 views

Condition for a product of two function sequences in $L^1$ to be in $L^1$

We have: 1. $\{f_n\} \subset L^1(E, \Sigma, \mu)$ 2. $g_n \subset L^\infty(E, \Sigma, \mu)$ 3. $\|f_n-f\|_{L^1}\to0$ 4. $g_n \to g \text{ }\mu\text{-a.e.}$ 5. $\{g_n\}$ is uniformly bounded. ...
0
votes
1answer
34 views

Converse of existing question on L^p convergence

My question is about this: Convergence in $L^p$ of $f_n$ implies convergence in $L^1$ of $|f_n|^p$ and $f_n^p$ It was shown that the author's question was indeed true by the use of MVT. Is the ...
3
votes
1answer
98 views

Convergence of characteristic functions on hypercube

I have a question regarding the following partition of a hypercube $H_{R}(x)$ centered at $x$ with sides of length $R$ in $\mathbb{R}^{n}$: Consider this hypercube $O = H_{R}(x) = ...
0
votes
0answers
9 views

Existence of uniformly continuous function on $L^p$

Suppose $f\in L^\infty(\Bbb R)$, and $f_h(x)=f(x+h)$ and $\lim_{h \to 0}||f_h-f||_\infty=0$. Prove that there exists a uniformly continuous function $g$ on $\Bbb R$ such that $f=g$ a.e. Problem is ...
0
votes
0answers
44 views

limit of p norm as p goes to 0!

Suppose we have a measure $\mu$ and a space $X$ such that $\mu(X)=1$, and a function $f \in L^r$ for some $r > 0$, where $L^r$ is defined in the usual way even for numbers less than $1$. Show ...
4
votes
1answer
55 views

$L^p$-space inclusions

Let $1\leq p<q<\infty$. Which of the following inclusions are true? $L^p(0,1)\subset L^q(0,1)$ $L^q(0,1)\subset L^p(0,1)$ $L^p(0,\infty)\subset L^q(0,\infty)$ $L^q(0,\infty)\subset ...
1
vote
1answer
44 views

$f_n\rightarrow f$ in $L_2$ and $f_n\rightarrow f$ in measure, then $f_n\rightarrow f$ almost uniformly?

$f_n\rightarrow f$ in $L_2$ and $f_n\rightarrow f$ in measure How to show or give an counterexample: $f_n\rightarrow f$ almost uniformly. We believe it is false. Since both convergences imply there ...
1
vote
1answer
37 views

What are the consequences of this simple property of $L^1$ functions?

I came across the following statement: Let $f\in L^1(\mathbb R,\mathbb R)$. Then $$\forall \varepsilon>0 \ \ \exists \delta>0 \ \ \text{such that for all open sets } U\subset\mathbb R \text{ ...
1
vote
2answers
77 views

What does it mean to be an L^1 function?

I am struggling to understand what the space L^1 is, and what it means for a function to be L^1. A friend told me that a function f is $L^1$ if $\int_\mathbb{R} |f|$ is finite. It is $L^2$ if ...
4
votes
3answers
71 views

How to apply the Hölder's inequality in a clever way?

Here is the problem: Let $f\in L^p(\mathbb R^n)\cap L^q(\mathbb R^n)$ and $s\in[p,q]$. Show that $f\in L^s(\mathbb R^n)$ I'm almost sure that this is a simple exercise on Hölder's inequality yet ...
1
vote
1answer
23 views

When is this function in Lp?

Trying to determine when $f(x)=|x|^{-\lambda}\in W^{1,p}(B)$ where $B\subset\mathbb{R}^n$ is the unit ball and $\lambda >0$. I've computed the distributional derivatives as $\partial_i ...