3
votes
1answer
40 views

Local convexity of the topology of weak convergence of probability measures

Let $X$ be a Polish or standard Borel space, and $\mathcal P(X)$ be the space of all Borel probability measures on $X$ endowed with the topology of weak convergence. I am thinking of using ...
1
vote
1answer
76 views

Strictly convex unit balls in $L^p$

I need to show that if $1<p<\infty$, then the unit ball is strictly convex in $L^p$, that is, $||\lambda x+(1-\lambda)y|| < 1$ whenever $||f|| = ||g||=1$ and $\lambda \in (0,1)$. I tried ...
0
votes
1answer
26 views

Certain property of convex functions…

I come to you with yet another qualifying problem we can't seem to solve... Let $f:$ $(0,\infty) \to \Bbb R$ be convex, and let $\lim_{x \to 0}f(x)=0$. Show that $g(x)$ = $f(x) \over x$ is increasing ...
1
vote
1answer
66 views

How to use Markov-Kakutani fixed point theorem to show that abelian groups are amenable?

Recall that a group $\Gamma$ is amenable if there exists a state $\mu$ on $l^{\infty}(\Gamma)$ which is invariant under the left translation action: for all $s\in \Gamma$ and $f\in ...
0
votes
0answers
21 views

Show that the graph of a convex function is above any tangent plane

In proving jensen inequality one use that the graph of a convex function is above any tangent plane. I've been reading Property of convex functions and Tangent line of a convex function. But what ...
1
vote
0answers
65 views

Dominated Convergence on risk measures

This is a quite specific question and I am not able to provide the whole background (e.g. what a risk measure is). If someone knows that would be great. I am having difficulties understanding a ...
1
vote
2answers
68 views

Integral of convex set

Let $D$ be a convex set and $X_1,\dots,X_d$ be integrable random variables. If $X= (X_1,\dots,X_d)$ is in $D$ almost surely, why is it true that the vector $a= (\mathbb{E}X_1,\dots,\mathbb{E}X_d)$ ...
0
votes
1answer
52 views

Measure of a set

Let $A \subset R$ such that for all open interval I, $m^* (A \cap I) < 1/2 L(I)$, where L is the length of a interval and $m^*$ is measure, prove that $m^*(A)=0$. I appreciate any hint to solve ...
3
votes
0answers
83 views

Averages of a Function

Let $(X,M,\mu)$ a measure space and $f:X \rightarrow \mathbb{C}$ a function in $L^{\infty}(\mu)$. Define $A_f$ as the set of all averages \begin{equation} \frac{1}{\mu(E)} \int_{E} f d \mu ...
5
votes
1answer
145 views

Taking a convex hull does not increase a supremum of a linear function

Let $X$ be a topological vector space, let $f:X\to\Bbb R$ be a continuous linear function and let $P(X)$ denote the set of all Borel probability measures on $X$. For any $M\subseteq X$ we define the ...
2
votes
2answers
231 views

An Orlicz norm is a norm

I had asked a question pertaining to Orlicz norms here. However, in the book I was reading, it said (and I paraphrase) "It's not difficult to show it is a norm on the space of integrable random ...
4
votes
1answer
203 views

strict convexity with a measure theoretic property

Suppose $(x_n)$ is a positive sequence of reals converging to $x$. Furthermore we have a measure space $(E,\mathcal{E},\mu)$ given, with finite measure $\mu$. There are a measurables nonnegative and ...
1
vote
0answers
58 views

A query about convexity in $L^p$ spaces

It defines the set $H^{p}_{\varepsilon}=\lbrace f \in L^{p}(0,1):\Vert f\Vert _{p}=(\int \vert f\vert^{p}dm)^{\frac{1}{p}}<\varepsilon\rbrace$ with respect to the measuring space ...
1
vote
1answer
141 views

Jensen's inequality and a estimate in $L^p$

In problem 3 we have: If $f:\mathbb{R} \longrightarrow\mathbb{R}$ is mensurable, $E:=\mathrm{supp}\ f$ and $$\int_E e^{|f(x)|}dx =1,$$ then $f\in L^p(\mathbb{R})$, for all $p\in(0,\infty)$ and ...
7
votes
1answer
285 views

Making sense out of “field”, “algebra”, “ring” and “semi-ring” in names of set systems

There are some set systems with algebraic titles, such as "field", "algebra", "ring" and "semi-ring" (and possibly other titles), in their names. Examples are a sigma field (aka sigma algebra, ...
5
votes
1answer
219 views

When does it make sense to define a base of a set system?

In a topology, a base is defined to be a class of subsets such that every open set is the union of some members of it. In a convexity structure, a base is defined to be a class of subsets ...
2
votes
2answers
119 views

When does it make sense to define a generator of a set system?

In a set system, such as a topology, sigma algebra or convexity structure, a generator is defined to be a class of subsets such that the given set system is the coarsest such set system ...
3
votes
1answer
132 views

Is it possible to replace function by its concave envelope

Let $f(x) \in C[-1,2]$. Consider an optimization problem $$ J[\mu] = \int\limits_{-1}^{2}f(x) \, \mu(dx) \to \max\limits_{\mu - \text{Borel probability measure}} $$ with restriction $$ ...
4
votes
0answers
80 views

Find optimal measure

Let $\Omega$ be a convex compact set in $\mathbb{R}^n$, $f\colon \Omega \to \mathbb{R}$ be a convex function. Consider an optimization problem $$ \int\limits_{\Omega}f(x)\,\mu(dx) \to ...
1
vote
3answers
112 views

Integral of exponential function

Consider $f$ being a measurable function on $R^n$ such that $$\int_{E} e^{|f|}=1$$ ($E$ measurable) and $f$ vanishes outside $E$ . Then $f\in L^p(R^n)$ for all $p\in (0,\infty)$. I tried using that ...
2
votes
0answers
183 views

$p$-norm is a convex function of $p$

For what measures $\mu$ and what intervals $(a,b) \subset (1,\infty)$ is the function $$p\mapsto \|f\|_p =\left(\int |f|^p d\mu \right)^{2/p}$$ a convex function of $p$ on $(a,b)$ for all $f\in ...
0
votes
1answer
62 views

Show $f(a)=\operatorname{E} \min(aX,Y)$ is a quasilinear function of $a$

Let $X$ and $Y$ be bounded real-valued random variables. Define $$ f(a)=\operatorname{E} \min(aX,Y) $$ Is $f$ a quasilinear function of $a$? That is $f$ is both quasiconvex and quasiconcave. To ...
1
vote
0answers
35 views

Uniqueness of function representation as a mean value

Let $f(x)$ be a smooth function from $\mathbb{R}^n_{+}$ to $\mathbb{R}_{+}$ (it may be analytic). Are there some sufficient conditions on such function $f$ such that $$ f(x_1 y_1, \ldots, x_n y_n) ...