Articles of measure theory

Volume form and Hausdorff measure

Let $M$ be a smooth orientable $(n-1)$-dimensional submanifold in $\mathbb{R}^n$, $dS$ be its volume form and $dH^{n-1}(x)$ be an $(n-1)$-dimensional Hausdorff measure. How to show than that $$ \int\limits_{M} f(x) dS = \int\limits_{M} f(x) dH^{n-1}(x) $$ In fact, it is a generaliation of an equality formula for surface integrals of first and second kind in […]

Can anyone give an example of a closed set contains no interval but with finite non-zero Lebesgue measure?

Can anyone give an example of a closed set $F$ of $\Bbb{R}$ such that $0<|F|<+\infty$ and $F$ contains no open interval? Thank you!

Does convergence in probability imply a.s. convergence in a countable space?

Let $(\Omega, \mathcal F,\mathbb P)$ be such that $\Omega$ is countable. I’m trying to find a simple example of random variables $X_n$ which converge to $0$ in probability but not a.s. If $\mathcal F = 2^{\Omega}$ (i.e., $\{\omega\} \in \mathcal F$ for each $\omega$ since $\Omega$ is countable), it is shown here that such random […]

Proof: $X\ge 0, r>0\Rightarrow E(X^r)=r\int_0^{\infty}x^{r-1}P(X>x)dx$

As the title states, the problem at hand is proving the following: $X\ge 0, r>0\Rightarrow E(X^r)=r\int_0^{\infty}x^{r-1}P(X>x)dx$ Attempt/thoughts on a solution I am guessing this is an application of Fubini’s Theorem, but wouldn’t that require writing $P(X>x)$ as an expectation? If so, how is this accomplished? Thoughts and help are appreciated.

Measure of countable union

In my measure theory class, the professor prove that if $\mu$ is a finite measure on a space $X$ ($\mu(X) < \infty$) and $A_1 \subset A_2 \subset A_3 \subset \cdots$, then $\lim_{n\to\infty} \mu(A_n)=\mu(\bigcup_{n} A_n)$. My question is: Is this false for infinite measures? I’ve tried to play a bit with the Lebesgue measure in $\mathbb{R}$, […]

Absolute continuity of a distribution function

This appeared on an exam I took. $Z \sim \text{Uniform}[0, 2\pi]$, and $X = \cos Z$ and $Y = \sin Z$. Let $F_{XY}$ denote the joint distribution function of $X$ and $Y$. Calculate $\mathbb{P}\left[X+ Y \leq 1\right]$. So this was easy – $$\begin{align} \mathbb{P}\left[X+Y \leq 1\right] &= \mathbb{P}\left[\sin Z+ \cos Z \leq 1\right] \\ &=\mathbb{P}\left[\sqrt{2}\sin\left(Z+\frac{\pi}{4}\right)\leq […]

Under what condition can converge in $L^1$ imply converge a.e.?

Let $f_n$ be a sequence of Lebesgue measurable functions on $R^d$. Suppose you have an estimate of the form $\int_{R^d}\left|f_n\right|\le c_n$ where $c_n \downarrow 0$. Can you conclude that $f_n\to 0$ a.e.? If not, what additional conditions on ${c_n}$ would guarantee this? My attempt: I think we cannot conclude that $f_n\to 0$ a.e. For example […]

How to show that the Dini Derivatives of a measurable function is measurable?

Let $f:(0,1)\to\mathbb R$ be measurable. Then, the (right upper) Dini derivative $$ D^+ f(x) = \limsup_{h\to 0^+} \frac{f(x+h) – f(x)}{h} $$ is also measurable (a well known result of Banach). Can someone give me a source in English (or German) or a proof sketch? If it makes thing much easier, we can assume $f$ is […]

What does “sets of arbitrarily large measure” mean — question about $L_p$ embeddings

The following are the results from a wikipedia article about $L_p$ space: a. Let $0 ≤ p < q ≤ ∞$. $L_q(S, μ)$ is contained in $L_p(S, μ)$ iff $S$ does not contain sets of arbitrarily large measure; b. Let $0 ≤ p < q ≤ ∞$. $L_p(S, μ)$ is contained in $L_q(S, μ)$ iff […]

Show: $\mathbb{E}(f|\mathcal{F})=\mathbb{E}(f)$

Let $(\Omega,\mathcal{A},\mathbb{P})$ be a probability space and $\mathcal{F}$ a sub-$\sigma$-algebra. Let $\mathcal{F}$ be trivial, i.e. $\forall A\in\mathcal{F}: \mathbb{P}(A)\in\left\{0,1\right\}$. Show that $\mathbb{E}(f|\mathcal{F})=\mathbb{E}(f)$. One criterion to prove that is to show that $$ \forall A\in\mathcal{F}: \int_A\mathbb{E}(f)\, d\mathbb{P}=\int_Af\, d\mathbb{P}. $$ Do not know exactly how to show that in common. For the special case, that $\mathcal{F}=\left\{\Omega,\emptyset\right\}$ it is […]