# Tag Info

### Proof of Krylov-Bogoliubov theorem

If you still need a reference: You can find a clear exposition of the result and the proof of the theorem in "Ergodic Theory" by Einsiedler 2013 (free PDF here). Or see the original work by ...
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Accepted

### Supremum of infimum of measure of members of a free ultrafilter

The answer is: zero. The reason is that every ultrafilter has zero as the infimum of the upper density of its members. To see this, observe that if a set $U$ is in the ultrafilter $\mathcal{U}$, with ...

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1 vote

1 vote
Accepted

• 85.1k
Accepted

### On conditions for the existence of $h\in L^1$ such that $h>0$ a.e

It is false without assuming that the measure is $\sigma$-finite. Let $\mu$ be the counting measure on $\mathbb{R}$. Then $h>0$ a.e. means $h>0$ everywhere and clearly  \int_{\mathbb{R}}hd\mu=\...
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### Representing an $L^2$-functional by a non-$L^2$-function on a dense subspace

I believe the following is a simple counterexample: Let $(X,\mu)$ be $\mathbb{N}$ with the counting measure (so I will be writing $\ell^2$ for $L^2(X,\mu)$. Let $g=0$ and $\tilde g(n) = (-1)^n$. Let ...
• 23.9k
Accepted

### Second order differentiability of convex functions

The following argument is used in my recent paper with D. Azagra and A. Cappello (work in progress). The answer is yes. I asked about whether the proof of Theorem 1 can be modified so to cover Theorem ...
• 23.9k
Accepted

### Is $L_p(X, \mu, E)$ uniformly convex for $p \in (1, \infty)$ if $E$ is a uniformly convex Banach space?

A reference for this result would be Some more uniformly convex spaces by Mahlon M. Day, Bull. Amer. Math. Soc. 47(6): 504-507 (June 1941). (Alternative link at Project Euclid)
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• 85.1k
### Are there some conditions on a metric space $X$ such that these two types of weak converge of finite signed Borel measures on $X$ are related?
An example. $X = [0,1]$ with the usual metric. $\mathcal C[0,1] = \mathcal C_b[0,1] = \mathcal C_0[0,1]$. $\mathcal C[0,1]^* = \mathcal M[0,1]$. Let $\mu_n$ be the unit point-mass at $1/n$ and $\mu$...