1
vote
3answers
53 views

Positive integral everywhere implies positive function a.e

I would like to get feedback on my demonstration of this simple statement : Let $f$ be an integrable function on the measure space $(X,S,\mu)$. \begin{align} \text{If }\int_E f \, d\mu \geq 0\text{ ...
1
vote
1answer
32 views

Triangle inequality for integrals with complex valued integrand

This is a step in a lecture note I'm reading. It should be simple because the author considers it obvious but I can't see it. What am I missing? Suppose $U$ and $V$ are integrable over measure space ...
1
vote
0answers
51 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
34 views

inequalities concerning integration and measure

Let $f$ be a non-negative function on $\mathbb{R}^n$ such that $\int_{\mathbb{R}^n} f=1$. Let $p\in(0,1)$. Let $E$ be any measurable subset of $\mathbb{R}^n$. Prove that $$ \int _E f^p\leq ...
1
vote
1answer
49 views

Generalized inverse of the cdf applied to a random variable equals the random variable itself almost surely?

first of all I apologize for the awful title but I really did not know how to formulate a precise question. Consider the following setup. Let $F$ be the distribution function of a random variable ...
0
votes
1answer
25 views

Application of the Hoelder inequality

How to prove using Hoelder Inequality that, $$\sum\limits_{i=1}^n \mathbb{E} (O (|X_i| + |X_i|^3 )) \leq n \, \mathbb{E} (O ( |X|^3 ) ),$$ where $X = (X_1, X_2, \ldots X_n)$ are i.i.d. independent ...
1
vote
1answer
42 views

One inequality involving Total variation function

If $F$ is of bounded variation in $[a,b]$, then I need to prove that $$ \int_{a}^{b}|F'(x)| dx \leq T_F(a,b)$$ If $F'$ were Riemann integrable then it was easy to prove (in fact we can prove ...
2
votes
0answers
42 views

Integral inequality related to derivation

While trying to understand a proof, i have stumbled upon the following statement: Let $f \in L^p(a,b)$ be a $p$-integrable function. Then the inequality $$\liminf_{s \rightarrow t} \frac{1}{t-s} ...
0
votes
2answers
54 views

Inequality with moments

Let $m$ a probability measure, $f$ a positive measurable function (one can assume it is bounded, the existence of the moments is not a problem here). Is $m(f^3) \le m(f^2) m(f)$?
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 ...
0
votes
1answer
54 views

Caratheodory's theorem and outer measure

I'm trying to show that $$\lambda(A)=\lambda(A\cap E)+\lambda(A\cap E^c)$$ where $\lambda$ is an outer measure, $A\subset \mathbb{R}$, $E \subset \mathbb{R}$, and $E$ is an elementary set; that is, ...
2
votes
1answer
56 views

Below bound of the mesure of a finite intersection

Let $(X, \mathcal{M}, \mu)$ be a measure space, with $\mu(X)=1$. If $A_{1}, A_{2}, ..., A_{n} \in \mathcal{M}$, prove that $$\mu \left(\bigcap_{j=1}^{n} A_{j} \right) \geq \sum_{j=1}^{n} \mu{(A_{j})} ...
4
votes
2answers
75 views

Analysis/Inequality question about proving an infinite product greater than 0

This is from David Williams' book Probability using Martingales. I'm self-studying. Question Prove that if $$0\leq p_n < 1 \quad\text{ and }\quad S:=\sum p_n < \infty$$ then $$\prod (1-p_n) ...
2
votes
1answer
210 views

Generalized Hölder inequality, the case when equality holds

I know the generalized Hölder inequality sounds like as: Let $1\leq p_1,\ldots,p_n<\infty$ and $p>0$ such that $\frac1p=\frac1{p_1}+\cdots+\frac1{p_n}$. Then, for all measurable functions ...
2
votes
1answer
39 views

Find $\inf_{f > 0} T_f := \left(\int_A f \, d\mu\right)\left(\int_A \frac{1}{f} \, d\mu\right)$

This exercise gives me trouble: Let $F$ denote the collection of measurable functions which are positive $\mu$-a.e. and let $A \in \mathbb X$ satisfy $0 < \mu(A) < \infty$. For $f \in F$ let ...
0
votes
1answer
75 views

best constant in quasi triangle inequality for $L^p$ spaces with $0 < p \le 1$

Currently doing a problem that ask me to prove the best $K$ such that the quasi triangle inequality $||f+ g||_p \le K (||f||_p + ||g||_p)$ for $L^p $ spaces holds, where $0< q \le 1$, is $2^{1/p -1 ...
0
votes
1answer
47 views

Some questions on the proof of Hoelders inequality.

I have some questions about the proof of Hoelder's inequality. Statement: Let $(X, \mathbb X, \mu)$ be a measure space. Let $p,q > 1$ with $1/p+1/q = 1$ and suppose that $f \in L_p(X)$ and $g \in ...
1
vote
1answer
103 views

Absolute value bound of Lebesgue integral

For the Riemann integral, we have the bound $$\left|\int_Af(x)dx\right|\leq\left(\sup_{x\in A}|f(x)|\right)\cdot\left|\int_Adx\right|$$ Do we have a similar bound for the Lebesgue integral, one like ...
0
votes
1answer
61 views

Proof of the nonexistence of an identity $\phi$ involving convolution

The Banach space $L^1(\mathbb{R}^n)$ is an algebra with a product (convolution) which is both commutative and associative. But this algebra does not have a multiplicative identity. An attempt to show ...
3
votes
1answer
72 views

Interesting inequality $\|F\|_p\le \frac{\pi}{\sin(\pi/p)}\|f\|_p$ over $L^p$

Consider the function $$F(x)=\int_0^\infty \frac{f(y)}{x+y} \, dy, \quad0<x<\infty$$ Prove that if $1<p<\infty$, $$\|F\|_p\le \frac{\pi}{\sin(\pi/p)}\|f\|_p$$ and show that the constant ...
1
vote
0answers
45 views

Intuition behind Gaussian isoperimetric inequality

I was wondering whether or not there's an intuitive way of understanding the Gaussian isoperimetric inequality. I have been studying the Classical isoperimetric inequality and I finally understand it. ...
2
votes
1answer
75 views

Esséen concentration inequality

I want to prove the following: Let $X$ be a random variable taking values in $\mathbb{R}^d$. Then for any $r>0, \epsilon >0$: $\displaystyle \sup_{x_{0}\in \mathbb{R}^d}{\bf{P}}(|X-x_{0}|\leq ...
4
votes
3answers
90 views

Prove that $λ^∗(A×B)\geq λ^∗(A)λ^∗(B)$ for every pair of sets, $A \subseteq \mathbb{R}^n$ and $B \subseteq \mathbb{R}^m$

Prove that $ \lambda^*(A\times B)\geq \lambda^*(A) \lambda^*(B)$ for every pair of sets, $A \subseteq\mathbb{R}^n$ and $B \subseteq\mathbb{R}^m$, where $\lambda^*$ denotes the Lebesgue Outer Measure ...
6
votes
0answers
370 views

Azuma's inequality to McDiarmid's inequality?

I was going through some notes on concentration inequalities when I noticed that there are two commonly-cited forms of McDiarmid's inequality. Long story short: I know how to prove the weaker one from ...
3
votes
0answers
150 views

Chebyshev and Markov inequalities

Chebyshev inequality: Let $(\mathcal{X},\mathcal{A},\mu)$ be a measurable space, $f$ a non-negative measurable function defined on $\mathcal{X}$. Then, $$\mu([f>c]) \le \frac{1}{c^p} ...
3
votes
0answers
88 views

A question about the stability of a property of the normal distribution

Recall that the standard normal distribution can be characterized as the unique standardized (having mean zero and unit variance) distribution $P$ on $\mathbb{R}$ with the property that with $X$, $Y$ ...
1
vote
1answer
51 views

Inequalities for bounded lipschitz functions

Suppose $X$ is a metric space with metric $d$. Define $\lVert f(x)\rVert_\infty = \sup_x |f(x)|$ and $\lVert f(x)\rVert_{LIP}=\sup\{\frac{|f(x)-f(y)|}{d(x,y}:x\not=y\}$, let $\lVert ...
3
votes
1answer
46 views

Inequality between product measure and its projection

$\newcommand{\smin}{\setminus} \newcommand{\sset}{\subseteq}$If $\mu$ is a measure on $X$, $ \nu $ a measure on $Y, \gamma $ a measure on $X \times Y$ s.t. $ \gamma(A \times Y) = \mu (A) $ and ...
2
votes
1answer
33 views

Equality of two classes of measures

Let $\alpha \in (0,1)$ and $M_{\alpha}$ be a class of Borel nonnegative measures $\mu$ on $\mathbb{R}^n_{+}$ such that there exists $r(\mu) > 0$: $$ \int\limits_{\mathbb{R}^{n}_{+}} ...
3
votes
2answers
104 views

An inequality about sequences in a $\sigma$-algebra

Let $(X,\mathbb X,\mu)$ be a measure space and let $(E_n)$ be a sequence in $\mathbb X$. Show that $$\mu(\lim\inf E_n)\leq\lim\inf\mu(E_n).$$ I am quite sure I need to use the following lemma. ...
18
votes
1answer
652 views

Do inequalities that hold for infinite sums hold for integrals too?

Let $\mathbb{R}_{\geq0}$ denote the set of non-negative reals and $+\infty$, and $\mathbb{Z}^+$ denote the set of positive integers. I will also let $\lambda$ denote the Lebesgue measure on ...
2
votes
1answer
85 views

An inequality $\frac{(1-\lambda)^{2}}{4}\leq m\{x\in [0,1]: |f(x)|\geq \frac{\lambda}{2}\}$

In this question $f$ is a Lebesgue measurable function on $[0,1]$ with the property that $\|f\|_{2}=1,\|f\|_{1}=1/2$. I am trying to prove that $$ \frac{(1-\lambda)^{2}}{4}\leq m\left\{x\in [0,1]: ...
0
votes
1answer
66 views

How can I show this inequality?

Let $\lambda B=\{x\in\mathbb{R}^n:\ \|x\|<\lambda\}$. Let $\eta>0$, $r_n\in (0,\eta)$ and $r_n\rightarrow \eta$. Suppose $u$ is a measurable function defined in $\eta B$. How can i show that ...
5
votes
0answers
332 views

Equality condition in Minkowski's inequality for $L^{\infty}$

I am trying to find out when equality holds in Minkowski's inequality for $L^{\infty}$ (i.e. a necessary and sufficient condition for equality). I did a search and there was a discussion for the case ...
2
votes
1answer
87 views

How can I give a bound on the $L^2$ norm of this function?

I came across this question in an old qualifying exam, but I am stumped on how to approach it: For $f\in L^p((1,\infty), m)$ ($m$ is the Lebesgue measure), $2<p<4$, let $$(Vf)(x) = ...
1
vote
1answer
71 views

$|\mu+\nu|(E)\le|\mu|(E)+|\nu|(E)$

$\mu$ and $\nu$ are complex measure, and $|\mu|$ is the total variation, that is, $$|\mu|(E):=\sup\left\{\sum_{i=1}^\infty|\mu(E_i)|, \{E_i\}_{i=1}^{+\infty}\mbox{ is partition of }E\right\}.$$ Is ...
4
votes
1answer
548 views

Liapunov's Inequality for $L_p$ spaces

Let $1 \leq p,q < \infty$ and $0 \leq \lambda \leq 1$. If $r = \lambda p + (1 - \lambda)q$ and $f \in L_p \cap L_q $, then $$||f||_r^r \leq ||f||_p^{\lambda p} ||f||_q^{(1 - \lambda)q} \tag{*}$$ ...
1
vote
1answer
64 views

Inequality between two expectations!

I normally have problem with proving inequalities since there are many different inequalities and I'm usually confused on how to choose a proper one and focus on that to get my problem solved. Here is ...
6
votes
1answer
856 views

Hölder inequality from Jensen inequality

I'm taking a course in Analysis in which the following exercise was given. Exercise Let $(\Omega, \mathcal{F}, \mu)$ be a probability space. Let $f\ge 0$ be a measurable function. Using Jensen's ...
2
votes
2answers
362 views

Question from Rudin: Jensen?

I came across this question from Rudin's Real and Complex Analysis, 3rd Edition (p.75 # 25) "Suppose that $\mu$ is a positive measure on $X$ and $f:X\rightarrow (0,\infty)$ satisfies $\int_X f ...
7
votes
5answers
202 views

Proving an estimate for this integral

How can I show that$$\sqrt[3]6>\int_1^\infty\frac{(1+x)^{1/3}}{x^2}\mathrm dx?$$
0
votes
1answer
55 views

Double Integral Claim

How can I prove the following inequality: (where $f $ is nice enough) - Given a function $ f(x,y) : \Omega_1 \times \Omega_2 \to \mathbb{R} $ , and $\alpha,C_1,C_2 $ are some constants, ( $\Omega_i$ ...
2
votes
1answer
176 views

Poincaré Inequality - Product Of Measures

I'm given two euclidean spaces $ \mathbb{R}_1 , \mathbb{R}_2 $ , with probability measures on them , that satisfy the Poincaré's inequality: $ \lambda^2 \int_{\mathbb{R}^k} |f - \int_{\mathbb{R}^k} f ...
2
votes
1answer
58 views

$a\mapsto \log\left(\lVert f\lVert_{1/a}\right)$ is a convex map

Proof for $a\in (0,1)$ that $a\to F(a)= \log\left(\lVert f\lVert_{1/a}\right)$ is a convex map, if $f\in L^p(X)$ for all $p\geq1$ in some measure space $(X,\mu)$. I'd like to prove that for all ...
2
votes
0answers
120 views

Inequality involving norms.

Suppose $p,q,r \in[1, \infty)$ and ${1\over r} = {1\over p} + {1\over q}$ . How can I use Minkowski's Inequality for prove below? $$||fg||_r \le ||f||_p||g||_q$$
4
votes
1answer
739 views

Generalization of Hölder's inequality

Assume $1<p_k< \infty$ for $k=1,\ldots,N$ , and $\displaystyle\sum^N_{k=1}\frac{1}{p_k} =1$. I want to prove that $$\left|\int_X f_1 f_2\cdots f_N\; d\mu \right| \le \lVert f_1\rVert_{p_1} ...
0
votes
1answer
1k views

Holder's inequality

Suppose that $f$ and $g$ are two non negative real valued functions defined on a measure space $(X,\mu)$. Let $0<p<\infty$. Holder's inequality says that $\int fg d\mu\le \|f\|_p \|g\|_q$ where ...
2
votes
1answer
140 views

Equality in the Isoperimetric Inequality

Stein and Shakarchi, in their book Real Analysis, the third volume of the Princeton Lectures in Analysis series, give a proof of the isoperimetric inequality for closed rectifiable curves in ...
1
vote
0answers
84 views

Markov-like inequality for functionals

Dear fellow mathematicians, The Markov inequality reads, for $(\Omega, \mathcal{F}, \mu)$ being a measure space, and $f$ a real valued function on $\Omega$ (you can also see Stein, Singular Integrals ...
14
votes
2answers
713 views

On the equality case of the Hölder and Minkowski inequalites

I'm following the book Measure and Integral of Richard L. Wheeden and Antoni Zygmund. This is the problem 4 of chapter 8. Consider $E\subseteq \mathbb{R}^n$ a measurable set. In the following all the ...