Articles of measure theory

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 […]

Measurability of supremum over measurable set

Consider a finite-valued function $f : \mathbb{R}^n \rightarrow \mathbb{R}$; and a closed-valued, measurable, set-valued mapping $S: \mathbb{R}^m \rightrightarrows \mathbb{R}^n$ . Measurability is intended with respect to a finite measure $m: \mathcal{B}(\mathbb{R}^m) \rightarrow [0,1]$, where $\mathcal{B}(\mathbb{R}^m)$ are the Borel sets. I am wondering if the following mapping is measurable as well. $$ x \mapsto \sup_{y \in […]

Measure zero on all Fat Cantor Sets

Let $F_n\subset [0,1]$ be a Fat Cantor Set (so that $[0,1]\setminus F_n$ is dense) of Lebesgue measure $1 – 1/n$, and let $F = \bigcup_n F_n$. Does there exist a probability measure $\mu$ on $[0,1]$ such that $\mu(F + x) = 0$ for all $x\in [0,1]$? Here $+$ is a cyclic shift, so $0.5 + […]

Is there anything special about a transforming a random variable according to its density/mass function?

Lets say that $X\sim p$, where $p:x\mapsto p(x)$ is either a pmf or a pdf. Does the following random variable possess any unique properties: $$Y:=p(X)$$ It seems like $E[Y]=\int f^2(x)dx$ is similar to the Entropy of $Y$, which is: $$ H(Y):=E[-\log(Y)]$$ It seems like we can always make this transformation due to the way random […]

Semi-partition or pre-partition

For a given space $X$ the partition is usually defined as a collection of sets $E_i$ such that $E_i\cap E_j = \emptyset$ for $j\neq i$ and $X = \bigcup\limits_i E_i$. Does anybody met the name for a collection of sets $F_i$ such that $F_i\cap F_j = \emptyset$ for $j\neq i$ but $X = \overline{\bigcup\limits_i F_i}$ […]

Proving a set is Lebesgue Measurable

This question already has an answer here: $E$ measurable set and $m(E\cap I)\le \frac{1}{2}m(I)$ for any open interval, prove $m(E) =0$ 2 answers

Questions about Fubini's theorem

I was wondering what theorem(s) makes possible exchanging the order of Lebesgue integrals, for instance, in the following example: $$\int\nolimits_0^1 \int_0^x \quad 1 \quad dy dx = \int_0^1 \int_y^1 \quad 1 \quad dx dy,$$ or more generally $$\int_0^1 \int_0^x \quad f(x,y) \quad dy dx = \int_0^1 \int_y^1 \quad f(x,y) \quad dx dy.$$ I am not […]

Ergodicity of tent map

The dynamical system $T:[0,1]\to [0,1]$ defined by $$T(x) = \begin{cases} 2x, & \text{for } 0\leq x\leq \frac{1}{2}\\ 2-2x, & \text{for } \frac{1}{2}\leq x\leq 1 \end{cases}$$ is called the tent map. Prove that $T$ is ergodic with respect to Lebesgue measure. My work: Measure preserving map is ergodic if for every T-invariant measurable set is $m(A)=1$ […]