Articles of measure theory

Fractional part of $n\alpha$ is equidistributed

Let $\alpha$ be an irrational number. Then the sequence $\{\{n\alpha\}\}$ is equidistributed. I am using the following definition of equidistribution. A sequence $\{a_i\}$ is equidistributed if $\frac1n\sum\limits_{i=1}^n f(a_i) \to \lambda(f):=\int_0^1 f(x)\,dx$ for all continuous $f:[0,1]\to \mathbb{R}_{\ge 0}$. In the proof I used the fact that if $T_g(x)=(g+x) \pmod 1$, then $\lambda(f\circ T_g)=\lambda(f)$ for all continuous […]

Measurable functions on product measures

Let $ (X,\mu) $ be a measure space, and consider $X \times X$ with the product measure $\mu \times \mu $. Consider two functions $f$ and $g$ defined on $X \times X$ such that: $f$ is measurable. For a.e. x, the function $y \to g(x,y)$ is measurable. The function $\int g(x,y) dy$ is a measurable […]

Independence of a function and integral of a function

I have a probability space $(\Omega, M, P)$ and non-negative integrable functions on $\Omega \times [0,1]$, $F_1(\omega, t)$ and $F_2(\omega, t)$. For each $t \in [0,1]$, we have that $F_1(\omega, t)$ and $F_2(\omega, s)$ are independent for any $s \in [0,1]$. Also $ \int_{\Omega} F_n(\omega, t) dP(\omega) = 1 $ for each $t$. Does it […]

Are Cumulative Distribution Functions measurable?

It is well-known that CDFs (Cumulative Distribution Functions) of one-dimensional random variables are Borel measurable. But does the same apply to CDFs of multi-dimensional random variables (rvecs)? It suffices, for my purposes, to consider finite dimensional rvecs. Relevant definitions Let $n$ be an integer $\geq 2$. Call a probability measure over the Borel field on […]

Real Analysis, Folland Problem 5.3.29 The Baire Category Theorem

The Baire Category Theorem – Let $X$ be a complete metric space a.) If $\{U_n\}_1^\infty$ is a sequence of open dense subsets of $X$, then $\bigcap_1^\infty U_n$ is dense in $X$. b.) $X$ is not a countable union of nowhere dense sets. The name for this theorem comes from Baire’s terminology for sets: If $X$ […]

Measurable functions and compositions

Let $(Y,S)$ a measurable space and $\phi :X\to Y$ any function where $X\neq \emptyset$. Suppose that $f:X\to\mathbb{R}$ is measurable over $(X,S’)$, where $S’=\phi^{-1}(S)$. I want to prove that there exists $g:Y\to\mathbb{R}$ measurable such that $f=g\circ\phi$. How can we define such $g$? Thanks!

Possible areas for convex regions partitioning a plane and containing each a vertex of a square lattice.

If the plane is partitioned into convex regions each of area $A$ and each containing a single vertex of a unit square lattice, is $A\in (0,\frac{1}{2})$ possible? If each each vertex is in the interior of its region is $A \neq 1$ possible? More generally if $\rm{ I\!R}^n$ ($n\ge 1$) is partitioned into convex regions, […]

How to show the following definition gives Wiener measure

On the first page of Ustunel’s lecture notes, he defines the Wiener measure in the following way: Let $W = C_0([0,1]), \omega \in W, t\in [0,1]$, define $W_t(\omega) = \omega(t)$. If we denote by $\mathcal{B}_t = \sigma\{W_s; s\leq t\}$, then there is one and only one measure $\mu$ on $W$ such that 1) $\mu \{W_0(\omega) […]

When does intersection of measure 0 implies interior-disjointness?

If there are two “nice” shapes in $R^2$, such as circles or polygons, whose intersection has area 0, then they must be interior-disjoint, as their intersection can only contain pieces of their boundary. My question is: what is a simple term for those “nice” subsets of $R^2$ for which intersection of area 0 implies interior-disjointness? […]

Is there always a Minimal Product Measure

I am studying measure theory and I have a question concerning the wikipedia-article “Product measure”. I already asked on the Wikipedia-“talk”-page but so far noone answered. The problem concerns the “minimal product measure”. I copy the following from the Wikipedia talk page: Take Omega to be any uncountable set. Take Sigma to be the power […]