Articles of measure theory

A question concerning Borel measurability and monotone functions

I came across the following exercise in my self-study: If $f: \mathbb{R} \rightarrow \mathbb{R}$ is monotone, then $f$ is Borel measurable. I am unsure about how to proceed from the hypothesis to give the requisite proof, in particular how sensitive I should be to proof by cases. Would anyone visiting have any suggestions, or be […]

Checking Caratheodory-measurability condition on sets of the semiring

Let $\mathcal H$ be a semiring over the set $X$ and $\mu$ a pre-measure defined on $\mathcal H$. Then we associate an outer measure $\mu^\ast$ to $\mu$ (describe here: (method I)) A set $A$ is called Caratheodory-measurable by $\mu^\ast$ if $\mu^\ast(Q) = \mu^\ast(Q \cap A) + \mu^\ast(Q \cap A^c)$ for all sets $Q \subset […]

Prove that $F(x,y)=f(x-y)$ is Borel measurable

Suppose $A$ is a subset of $\Bbb R$, let $s(A)=\{ (x,y)\in \Bbb R \times \Bbb R :x-y\in A\}$. I already showed: If $A\in \Bbb B$ (Borel measurable set), then $s(A)\in \Bbb B \times \Bbb B$. I want to use this to prove that if $f$ is a Borel measurable function on $\Bbb R$ to $\Bbb […]

The finite-dimensional distributions of a centered Gaussian process are uniquely determined by the covariance function

Let $I\subseteq\mathbb{R}$ and $X=(X_t)_{t\in I}$ be a centered Gaussian process, i.e. – $E[X_t]=0$ for all $t\ge 0$ – $X$ is real-valued and for all $n\in\mathbb{N}$ and $t_1,\ldots,t_n\ge 0$ we’ve got $$(X_{t_1},\ldots,X_{t_n})\;\;\;\text{is }n\text{-dimensionally normal distributed}$$ How can we prove, that the finite-dimensional distributions of $X$ are uniquely described by the covariance function $$\Gamma(s,t):=\operatorname{Cov}[X_s,X_t]\;\;\;\text{for }s,t\in I\;?$$

Lebesgue measure of any line in $\mathbb{R^2}$.

What is the Lebesgue measure of a line in $\mathbb R^2$? I am guessing that this zero. But i couldn’t prove it rigorously. Please help… From this can i conclude that any proper subspace of $\mathbb R^n$ has measure zero.

Uniform measure on the rationals between 0 and 1

I am trying to think of a probability measure on the set of rationals between 0 and 1 ($X:=\mathbb{Q}\cap[0,1]$). I want to achieve something like a uniform measure, i.e. every number should have the same probability. Of course from the fact, that there are infinitely many atoms, it follows, that the measure has to be […]

How to show $\mathcal{L}(\mathbb{R}) \otimes \mathcal{L}(\mathbb{R}) \subset \mathcal{L}(\mathbb{R^2})$?

Let $\mathcal{L}(\mathbb{R})$ be the lebesgue-measurable set of $\mathbb{R}$, and $\mathcal{L}(\mathbb{R^2})$ the lebesgue-measurable set of $\mathbb{R^2}$. First I shall show, that $\mathcal{L}(\mathbb{R}) \otimes \mathcal{L}(\mathbb{R})$ is a subset of $\mathcal{L}(\mathbb{R^2})$. In a second part I shall show, that in fact they are not equal. There is a hint, that the Lebesgue-measurable sets are the completion of the […]

Properties of Weak Convergence of Probability Measures on Product Spaces

EDIT: For the Bounty, I made a substantial edit revision concerning the structure of the question, to make it more readable (hopefully). Moreover I added a question on problem 2.7 of Billingsley’s book. I have two problems concerning weak convergence of probability measures in product spaces, that arose from Billingsley’s classic “Convergence of Probability Measures” […]

Normed Linear Space – maximum norm vs. $||f||_1$

For $f$ in $C[a,b]$ define $$|| f ||_1 =\int_a^b |f|.$$ a. Show that this is a norm on $C[a,b]$. b. Show that there is no number $c \geq0$ for which $$||f||_{max} \leq c ||f||_1 \ for \ all \ f \ in \ C[a,b]$$ c. Show there is a $c\geq 0$ for which $$||f||_1 \leq […]

Step Function and Simple Functions

Definitions: Simple Function: Any functions that can written in the form:$$s(x)=\sum_{k=1}^na_n\chi_{A_n}(x).$$ Note the finite terms here. It should follow that neither all simple functions are step functions, nor all step functions are simple function. e.g. Would not Cantor Function or Devil’s Staircase be example of step function but not simple (note again the finite)? I […]