Articles of measure theory

How to use the modular function of a locally compact group?

Let $G$ be a locally compact group with left Haar measure $\mu$. Let $\delta:G\to(0,\infty)$ the unique modular function such that $\delta(x)\mu(E) = \mu(Ex)$ for all Borel sets $E$ and $x\in G$. How is this definition used in integration? Are the following steps correct? (Please alert me of even the slightest mistake!) For fixed $x\in G$, […]

Does it make sense to talk about $L^2$ inner product of two functions not necessarily in $L^2$?

The $L^2$-inner product of two real functions $f$ and $g$ on a measure space $X$ with respect to the measure $\mu$ is given by $$ \langle f,g\rangle_{L^2} := \int_X fg d\mu, $$ When $f$ and $g$ are both in $L^2(X)$, $|\langle f,g\rangle_{L^2}|\leqslant \|f\|_{L^2} \|g\|_{L^2} < \infty$. I was wondering if it makes sense to talk […]

$L^p$ 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.