4
votes
2answers
42 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
64 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
40 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
41 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
58 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
43 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
24 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
1answer
32 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
42 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
22 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
91 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
33 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
45 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
37 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
63 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
32 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
48 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
40 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
63 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
46 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
30 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
74 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
6 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
38 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
47 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
42 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
72 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
70 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 ...
0
votes
2answers
31 views

1.Convergence in $L^1$

Let f be measurable function such that $||f||_\infty=\infty.$ Show that there exists {${g_ n}$} $\subset L^1$ such that $||fg_n||_1\to\infty$. Anyhelp would be appreciated..
0
votes
0answers
49 views

Inclusion of $L^p$ spaces if $X$ arbitrary

If $X$ is a finite measure space then one can show that if $1 \le p < q$ then $L^q \subseteq L^p$. Is there anything known about the inclusion if $X$ is an arbitrary measure space? Or given some ...
1
vote
1answer
22 views

Is a set of jointly bounded functions over a compact domain compact under p-norm?

Let $X$ be a metric space and a measurable space. Let $K$ be a compact set of nonzero measure and $r> 0$. Is a set $\{ f: K\rightarrow \mathbb R| |f|\leq r$ almost everywhere$\}$ compact with ...
6
votes
1answer
73 views

Proof of equivalence of Sobolev Space and Lipschitz functions

The attachment is a proof from Evans book "Measure Theory and Fine Properties of Functions" pg 132 Theorem 5. The statement of the theorem is: Let $f:U \rightarrow \mathbb{R}$. Then $f$ is locally ...
1
vote
0answers
45 views

Norm of multiplication operator

I have that $(X,\Omega,\mu)$ is a sigma finite space, and I have that $g$ is a measurable function. Assume that $fg\in L^p$ for all $1\leq p\leq \infty$. I want to show that $g\in L^\infty$. My idea ...
3
votes
1answer
79 views

Are $L_p$ spaces of functions with separable support separable?

Let $X$ be a separable space. Is $L_p$$(X, \mu, V)$ a separable space? Here, $(V, |\cdot|_V)$ is a normed space. And a norm of $L_p(X, \mu, V)$ is: $$ \|f\|=\left(\int_X \big(\vert f\rvert_V\big)^p ...
1
vote
1answer
36 views

What is the meaning of “$fg$ has constant sign a.e.”?

What is the meaning of "$f$g has constant sign a.e." for question 4 on page 143 of Wheeden & Zygmund's Measure and Integral?
1
vote
1answer
45 views

Equivalent condition for equi-integrability

I am looking for a Lemma that gives an equivalent formulation for a family of functions to be equi-integrable: is it true that if $\{f_j\}_j\in L^1$, then we can write $f_j=f^1_j+f_j^2\in L^1+L^p$, ...
3
votes
0answers
38 views

Relationship between complex and real Lp spaces.

The theory of integration of real functions is (as far as I know) usually extended to the complex case as follows: Let $X$ be a set. Given a function $f:X\to\mathbb C$, define ...
3
votes
1answer
115 views

If $f \in L^{\infty}$ and $\exists r < \infty$ so that $\|f\|_r < \infty$, show $\lim_{p \rightarrow \infty} \|f\|_p = \|f\|_{\infty}$ [duplicate]

Question: This is the last part of a 5 part question I am working on. Let $(X,\mu)$ be a possibly infinite measure space. Assume $\exists r < \infty$ with $\|f\|_r < \infty$ and that $\|f ...
1
vote
1answer
55 views

Approximation of bounded Borel functions

Let $K$ be a compact space and let $B$ be the space of bounded Borel functions on $K$ equipped with the supremum norm. Show that simple functions (i.e. functions attaining only a finite number of ...
6
votes
1answer
49 views

Under what conditions on $f$ does $\|f\|_r = \|f\|_s$ for $0 < r < s < \infty$.

Question: If $f$ is a complex measurable function on $X$, such that $\mu(X) = 1$, and $\|f\|_{\infty} \neq 0$ when can we say that $\|f\|_r = \|f||_s$ given $0 < r < s \le \infty$? What I ...
1
vote
1answer
41 views

$\mathbb L^1 +$ a.s. convergence of sequence $(X_n)$ does not imply $\sup(x_n)$ is integrable

I'm looking for a counterexample. The setting is this: Given an probability space $(\Omega,\mathbb{F},\mathbb{P}) $, I look for sequence of random variables $(X_n)_n$ and a random variable $X$, all in ...
1
vote
0answers
42 views

Convergence of $L^1$ functions

Given that $\Omega$ is bounded and $a_{ij}(u_{k}) \rightarrow a_{ij}(u)$ in $L^{1}(\Omega)$, $a_{i0}(u_{k}) \rightarrow a_{i0}(u)$ in $L^{1}(\Omega)$, $\frac{\partial u_{k}}{\partial x_{j}} ...
1
vote
0answers
43 views

$L\log L$ and $L^p$ embedding

My question is a simple one: I am aware of the embedding $L^p(\Omega)\in L\log L(\Omega)$ for finite measure spaces, with constant $\frac{cp}{p-1}$. Does this embedding hold on for instance, the whole ...
1
vote
1answer
57 views

Lp space over measure space and over its completion is the same.

It is exercise of Tao's 1.3.1 Let $(X, \chi,\mu)$ and $(X, \overline{\chi}, \mu)$ is a measure space, where $\overline{\chi}$ is the completion of $\chi$. Show that $L^{p}(X, \chi, \mu)$ is ...
0
votes
2answers
91 views

bounded linear functional on $\ell^{1}$, and its relation to $\ell^{\infty}$

Prove that a bounded linear functional $F$ on $\ell^1$ has representation $F(x)=\sum_{n=1}^{\infty}(c_{n}x_{n})$ where $c_{n} \in \ell^{\infty}$, and that $\|F\|_{*} = \|c_{n}\|_{\infty}$.
2
votes
1answer
61 views

Is $L^p \cap L^q$ dense in $L^r$?

It is known that $L^p \cap L^q \subset L^r$, where $1 \le p \le r \le q \le \infty$. Are all of these inclusions dense? I.e., do we have \begin{equation*} \overline{L^p \cap L^q} = L^r ...