Articles of measure theory

A simple curve of positive area

Let $\gamma : [0,1]\rightarrow\mathbb R^2$ be a continuous curve whose image has positive Lebesgue measure. Must $\gamma$ have self-intersections? Intuitively this seems like it should be true, but I could not find a proof. An equivalent question is whether there is a positive measure subset of $\mathbb R^2$ which is homeomorphic to $[0,1]$. Note that […]

When is $L^1 = (L^\infty)^\ast$?

I found this exercise in Cohn’s Measure Theory: Let $(X, \mathscr A, \mu)$ be a finite measure space. Show that the conditions the map $T: L^1(X, \mathscr A, \mu) \to (L^\infty(X, \mathscr A, \mu))^\ast$ given by $g\mapsto T_g(f) = \int fg \, d\mu$ is surjective $L^1(X, \mathscr A, \mu)$ is finite-dimensional $L^\infty(X, \mathscr A, \mu)$ […]

Does $\mathcal P ( \mathbb R ) \otimes \mathcal P ( \mathbb R ) = \mathcal P ( \mathbb R \times \mathbb R )$?

Obviously the answer’s yes if I replace $\mathbb R$ by a countable set. If $\mathbb R$ is replaced by a set $X$ having a greater cardinality, then the diagonal $\{ (x,x) , \ x\in X\}$ is not in the product $\sigma$-field (see for instance http://david.efnet-math.org/?p=16) so the answer is no.

Any lower semicontinuous function $f: X \to \mathbb{R}$ on a compact set $K \subseteq X$ attains a min on $K$.

I’ve been thinking about this problem for a long time right now, and feel stuck. Given that $X$ is a topological space, and that for $f$ to be lower semicontinuous, for any $x \in X$ and $\epsilon > 0$, there is a neighborhood of $x$ such that $f(x) – f(x') < \epsilon$ for all $x'$ […]

The Laplace transform of the first hitting time of Brownian motion

Let $B_t$ be the standard Brownian motion process, $a > 0$, and let $H_a = \inf \{ t : B_t > a \}$ be a stopping time. I want to show that the Laplace transform of $H_a$ is $$\mathbb{E}[\exp(-\lambda H_a)] = \exp (-\sqrt{2\lambda} H_a)$$ by considering the martingale $$M_t = \exp \left(\theta B_t -\frac{1}{2}\theta^2 t\right)$$ […]

If $f_k \to f$ a.e. and the $L^p$ norms converge, then $f_k \to f$ in $L^p$

Let $1\leq p < \infty$. Suppose that $\{f_k\} \subset L^p$ (the domain here does not necessarily have to be finite), $f_k \to f$ almost everywhere, and $\|f_k\|_{L^p} \to \|f\|_{L^p}$. Why is it the case that $$\|f_k – f\|_{L^p} \to 0$$ A statement in the other direction (i.e. $\|f_k – f\|_{L^p} \to 0 \Rightarrow \|f_k\|_{L^p} \to […]

Show that $\lim _{r \to 0} \|T_rf−f\|_{L_p} =0.$

I am having a hard time with the following real analysis qual problem. Any help would be awesome. Thanks. Suppose that $f \in L^p(\mathbb{R}),1\leq p< + \infty.$ Let $T_r(f)(t)=f(t−r).$ Show that $\lim_{r \to 0} \|T_rf−f\|_{L_p} =0.$

Let $E$ be measurable and define $f:E\rightarrow\mathbb{R}$ such that $\{x\in E : f(x)>c\}$ is measurable for all $c\in\mathbb{Q}$, is $f$ measurable?

Let $E$ be measurable and define $f:E\rightarrow\mathbb{R}$ such that $\{x\in E : f(x)>c\}$ is measurable for all $c\in\mathbb{Q}$, is $f$ measurable? There are a number of equivalent definitions for the measurability of a function and the most obvious one would be to show that $\{x\in E : f(x)>c\}$ is measurable for all $c\in\mathbb{R}$. Thus my […]

Axiom of choice, non-measurable sets, countable unions

I have been looking through several mathoverflow posts, especially these ones https://mathoverflow.net/questions/32720/non-borel-sets-without-axiom-of-choice , https://mathoverflow.net/questions/73902/axiom-of-choice-and-non-measurable-set and there still are many questions I would like to ask: 1) According to the first answer of the first post “It is consistent with ZF without choice that the reals are the countable union of countable sets” (and therefore all […]

Convolution of an $L_{p}(\mathbb{T})$ function $f$ with a term of a summability kernel $\{\phi_n\}$

… is the result in $L_{p}$? A remark in my notes says yes but I can’t see how to verify it. As was pointed out to me in a previous question I asked last night, I need to show that the following integral is finite: $$\int_{-\pi}^{\pi}|\int_{-\pi}^{\pi}f(t-s)\phi_{n}(s)ds|^{p}dt < \infty$$. One of the properties of a summability […]