Articles of measure theory

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 $m(A)=1$ […]

Show that for any $f\in L^1$ and $g \in L^p(\mathbb R)$, $\lVert f ∗ g\rVert_p \leqslant \lVert f\rVert_1\lVert g\rVert_p$.

I write the exact statement of the problem: Show that for any $g \in L^1$ and $f ∈ L^p(\mathbb{R})$, p $\in (1, \infty)$, the integral for f ∗g converges absolutely almost everywhere and that $∥f ∗ g∥_p ≤ ∥g∥_1∥f∥_p$. I’ve tried to show that the convolution is well defined for f,g $\in L^1$ in this […]

Do probability measures have to be the same if they agree on a generator of Borel $\sigma$–algebra $\mathcal{B}(\mathbb{R})$?

Suppose $\mathcal{K}\subset 2^\mathbb{R}$ is such that $\sigma(\mathcal{K})=\mathcal{B}(\mathbb{R})$ and let $\mu$ and $\nu$ be measures which agree on $\mathcal{K}$, i.e. $$\mu(A)=\nu(A)$$ for all $A\in\mathcal{K}.$ Do these measures necessarily have to be the same? I am teaching myself in measure-theoretic probability and for now I can only prove that the answer is yes, assuming $\mathcal{K}$ is a […]

Can locally “a.e. constant” function on a connected subset $U$ of $\mathbb{R}^n$ be constant a.e. in $U$?

Consider a non-empty connected open subset $U$ of $\mathbb{R}^n$. Suppose a measurable function $u:U\to\mathbb{R}$ is locally constant on $U$, then it must be constant on $U$ according to this question. Here is my question: What if one changes “locally constant” to “locally a.e. constant”? More precisely, assume that for every $x\in U$ there is an […]

On atomic and atomless subsets

In a measure space, let’s call a measurable subset atomless wrt the measure, if it does not have an atomic subset. In particular a measurable subset with zero measure is atomless. There may be measurable subsets that are neither atomic nor atomless, for instance, the union of atomic subset(s) and atomless subset(s) with positive measure(s). […]