# Tagged Questions

Questions related to measures, sigma-algebras, measure spaces, Lebesgue integration and the like.

35 views

### Lebesgue outer measure is countably subadditive but not finitely additive proof

I have read all the Qs on this but couldn't find a clear proof. How can I prove that Lebesgue's outer measure is not finitely additive? Thanks! Edit: I understand I must show that the measure of the ...
95 views

### convergence in distribution in Banach spaces

We let $\Omega$ be a compact metric space and consider $C(\Omega)$ to be the space of all continuous functions on $\Omega$. The dual space of $C(\Omega)$ can be seen as the set of all signed borel ...
46 views

### Real Analysis, Folland Proposition 2.29 Modes of Convergence

Background Information: $f_n\rightarrow f$ in $L^1$ $\Leftrightarrow$ $\forall\epsilon > 0,\exists N$ $\forall n\geq N$ $\int |f_n - f| < \epsilon$ A sequence $\{f_n\}$ of measurable complex-...
59 views

### If $X_n \stackrel{p, quickly}{\to} X$, then $X_n \to X$.

Probability with Martingales: Without using hint, can I just do something like this: http://math.stackexchange.com/a/1538503/140308 ? With using hint: By continuity of probability, I think ...
16 views

45 views

41 views

34 views

5 views

### Haar measure, can image of modular function be any subgroup of $(0,\infty)$?

It is easy to find examples of locally compact second countable Hausdorff topological groups $G$ whose modular function $\Delta$ has image $\{1\}$ or $(0,\infty)$. Are there groups $G$ of this kind ...
27 views

51 views

### Theorem 2.17 from RCA Rudin

I understood the proof of points $(a)$ and $(c)$. But I can't understand the proof of $(b)$. It's obvious that every closed set is $\sigma$-compact. But how Rudin applies $(a)$ here? We have to show ...
42 views

### Recent advancement in Haar measure

From my personal interest I have studied Haar Measure and the related concept of group theory on my own. However due to the lack of an authoritative source it is not getting possible for me to know ...
56 views

### Real Analysis, Folland problem 2.2.16 Integration of Nonnegative functions

If $f\in L^+$ and $\int f < \infty$, for every $\epsilon > 0$ there exists $E\in M$ such that $\mu(E) < \infty$ and $\int_E f > (\int f) - \epsilon$. Attempted proof - Let $f\in L^+$ and ...
54 views

### Real Analysis, Folland Problem 2.2.14 Integration of Nonnegative functions

Problem 2.2.14 - If $f\in L^{+}$, let $\lambda(E) = \int_{E}f d\mu$ for $E\in M$. Then $\lambda$ is a measure on $M$, and for any $g\in L^{+}$, $\int g d\lambda = \int f g d\mu$.(First suppose that $g$...
51 views

### $m_*(E)=m^*(E)\iff E$ Lebesgue measurable
Let $E\subset [a,b]$. Show that $E$ is Lebesgue measurable if and only if the Lebesgue outer measure of $E$ is equal to the Lebesgue inner measure of $E$. I have seen the proof for this above ...
It's Riesz Representation Theorem from Rudin's book. In the following chapter I met the following example: It's obvious that $\sigma$-compact set has the $\sigma$-finite measure. But how to prove ...