Articles of measure theory

$L^p$ norm and integral equality prove

Possible Duplicate: Integrate and measure problem. Assume $\mu(X)=1$, $f \in L^{p} (X,M,\mu)$ for some $0<p \le \infty$ I want to prove that: $$\lim_{p\to 0}||f||_p = e^{\int_X \log|f|d \mu}$$ I’m going to prove $\ge$ part using Jensen inequality, but I cannot go opposite side. How can I make it?

Riemann Integral Problem

I’m studying Riemann Integral on measure theory class. There is a function $f : [a,b] \to \mathbb{R}$ that is increasing or decreasing. Is that function f is Riemann integrable? And if then, what are appropriate step functions? And is that function continuous a.e.?

Let $\mu $ be a $\sigma $-finite measure. Show that for any $f\in L_p(\mu)$, $\|f\|_1=\sup\{ \int fg\, d\mu :\|g\|_\infty \leq 1\}$

Let $\mu $ be a $\sigma $-finite measure. Show that for any $f\in L_p(\mu)$, $\|f\|_1=\sup\{ \int fg \, d\mu :\|g\|_\infty \leq 1\}$ I know that Holders inequality implies $\int fg \, d\mu \leq \|f\|_1 \|g\|_\infty$ and for all $\|g\|_\infty \leq 1$, $\int fg \, d\mu \leq \|f\|_1$. But how do I show it’s the supremum? […]

Expectation of first-passage-time of a diffusion process with negative drift

Take the stochastic process $X_0 = 0$ and $X_t = \nu t + \sigma W_t$ where $W_t$ is standard Brownian motion and $\nu$ is a drift which may be negative (this is the key complexity to the question) Let $\alpha > 0$ be a fixed level, then define the first passage time as the random […]

$\sigma$- ideal

Let $(\Omega,\mathcal{A})$ be a measurable space. $\mathcal{N}\subset\mathcal{P}(\Omega)$ is called a $\sigma$ ideal, if $$ (1)~\emptyset\in\mathcal{N},~~~~~(2) N\in\mathcal{N}, M\subset N\Rightarrow M\in\mathcal{N},~~~~~(3)(N_n)\in\mathcal{N}^{\mathbb{N}}\Rightarrow\bigcup_n N_n\in\mathcal{N} $$ Show that for every $\sigma$-ideal $\mathcal{N}$ it is $$ \sigma(\mathcal{A}\cup\mathcal{N})=\left\{A\Delta N|A\in\mathcal{A},N\in\mathcal{N}\right\}. $$ Hint: $$ \left\{A\Delta N|A\in\mathcal{A},N\in\mathcal{N}\right\}=\left\{B\subset\Omega|\exists A\in\mathcal{A},N\in\mathcal{N}: B\setminus N=A\setminus N\right\} $$ I do not have a special idea, to be honest. For the inclusion […]

Is Lebesgue measure the only measure that is locally finite, strictly positive and translation-invariant on $\mathbb{R}^n$?

Lebesgue measure on Euclidean space $\mathbb{R}^n$ is locally finite, strictly positive and translation-invariant. Is Lebesgue measure the only such measure on $\mathbb{R}^n$? Thanks!

What is the Lebesgue measure of the set of numbers in $$ that has two thirds of ones in their infinite base-2 expansion?

Let $S= \{x \in [0,1 ]: \lim_{n \to \infty } \frac {1 } {n } \sum _{i=1 } ^n d _i (x) = 2/3 \} $, where $d _i(x) $ gives the ith digit of the infinite base-2 expansion of $x $. What is the Lebesgue measure of $S $? First I want to prove […]

The Radon–Nikodým theorem for vector valued measures

I am looking for a proof of the Radon–Nikodým theorem in the case of vector valued measures. Many textbooks cover the scalar case. The book I am reading mentions the vector valued case but does not provide a proof or a reference. Any help is greatly appreciated. Thanks, Phanindra.

Completion of a measure space

Studying Measure Theory in University, I came across the following definition for the completion of a measure space: let $(X,\mathcal{E},\mu)$ be a measure space; then the set $$\overline{\mathcal{E}}=\left\lbrace A\subseteq X:\exists B,C\in\mathcal{E}:A\triangle B\subseteq C\wedge\mu(C)=0\right\rbrace$$ is a $\sigma$-algebra and, extending $\mu$ to $\overline{\mu}$ defined on $\overline{\mathcal{E}}$ by letting $\overline{\mu}(A)=\mu(B)$ for any set $B\in\mathcal{E}$ such that $A\triangle B$ […]

Is the integral $\int_1^\infty\frac{x^{-a} – x^{-b}}{\log(x)}\,dx$ convergent?

Is the integral $$\int_1^\infty\frac{x^{-a} – x^{-b}}{\log(x)}\,dx$$ convergent, where $b>a>1$? I think the answer lies in defining a double integral with $yx^{(-y-1)}$ and applying Tonelli’s Theorem, but the integral of $\frac{x^{-a}}{\log x}$ is still not integrable. Any ideas?