Articles of measure theory

Breaking up a countable sum

Let $W$ be a countable set and $p:W\to[0,1]$ be any function satisfying $$ \sum_{w\in W}p(w)=1. $$ Now, let $A_1,A_2,\ldots$ be a countable collection of disjoint subsets of $W$. With $A=\bigcup_n A_n$, it is intuitive that $$ \sum_{w\in A}p(w)=\sum_n \sum_{w\in A_n}p(w). \tag{*} $$ Do I need to justify this seemingly obvious implication? If so, could you […]

How to understand marginal distribution

Given a random vector $X: (\Omega, \mathbb{F}, P) \rightarrow (\prod_{i \in I} S_i, \prod_{i \in I} \mathbb{S}_i)$, is each component variable $X_i, \forall i \in I$ of the random vector $X$ always a random variable from $(\Omega, \mathbb{F}, P)$ to $(S_i, \mathbb{S}_i)$? If yes, I guess there are two ways to define the distributions for […]

Jordan Measure, Semi-closed sets and Partitions

From earlier question here. Consider $$\mu \left( (0,1) \cap \mathbb Q \right) = 0$$ where $$(0,1) = \left(0,\frac{1}{3}\right)\cup \left[\frac{1}{3}, \frac{2}{3}\right] \cup \left(\frac{2}{3}, 1\right)$$ so $$ \mu ((0,1) \cap \mathbb Q ) = \mu \left(\left(0,\frac{1}{3}\right) \cap \mathbb Q \right) + \mu\left( \left[\frac{1}{3}, \frac{2}{3}\right] \cap \mathbb Q\right) + \mu\left(\left(\frac{2}{3}, 1\right) \cap \mathbb Q\right) = 0 + \frac{1}{3} […]

$\int^b_a t^kf(t) dt\,=0$ for all $k \geq 1 \implies f=0$ a.e.

Let $f\in L^1[a,b]$ satisfying $$\int^b_a t^kf(t) dt\,=0$$ for all positive integer $k$. Show that $f=0$ a.e. I did a similar problem where $\int^b_a t^kf(t) dt\,=0$ was true for all $k\in \Bbb{N} \cup \{0\}$. This is relatively easy and I did it using Weierstrass approximation theorem, but how to do when $k\in \Bbb{N}$ and k cannot […]

An equality concerning the Lebesgue integral

Let $f:X\mapsto[0,+\infty)$ be a non-negative measurable function defined on the space $X$, endowed with the complete $\sigma$-additive, $\sigma$-finite, measure $\mu$ defined on the $\sigma$-algebra of the measurable subsets of $X$. I have read that the following equality holds for the Lebesgue integral: $$\int_X f d\mu = \int_{[0,+\infty)} \mu(\{x\in X: f(x)>t\}) d\mu_t$$where $\mu_t$ is the usual […]

Definition of upper integral

Answering this question it occurred to me that the OP’s definition of integral is unsatisfactory in the following sense. He defines it using the usual Lebesgue integral. I think it would be far more satisfactory if we could define the integral without using Lebesgue integral. In other words, it would be far more satisfactory if […]

Compactness in $L^1$

Let $\mu(\cdot)$ be a probability measure in $X$. Consider a function $f: Z \times X \rightarrow \mathbb{R}_{\geq 0}$, with $Z \subset \mathbb{R}^n$ compact, such that: $\forall x \in X, \quad z \mapsto f(z,x)$ is continuous; $\forall z \in Z, \quad x \mapsto f(z,x)$ is measurable; The family $\mathcal{F} = \{ f(z,\cdot) \}_{z \in Z}$ is […]

Complex Functions: Integrability

Let $\Omega$ be a measure space with measure $\lambda$. Denote the space of simple functions by: $$\mathcal{S}:=\{s:\Omega\to\mathbb{C}:s=\sum_{k=1}^{K<\infty}s_k\chi_{A_k:\lambda(A_k)<\infty}\}$$ Denote the positive and negative part of the real and imaginary part by: $$f=\Re_+f-\Re_-f+i\Im_+f-i\Im_-f=:\sum_{\alpha=0\ldots3}i^\alpha f_\alpha$$ Define for positive functions: $$\int fd\lambda:=\sup_{s\in\mathcal{S}:s\leq f}\int sd\lambda\quad(f\geq 0)$$ and for complex functions: $$\int fd\lambda:=\sum_{\alpha=1\ldots3}i^\alpha\int f_\alpha d\alpha$$ as long as all terms of […]

Approximation of measurable function by simple functions

This is from Tao Exercise 19.1.3. Let $\Omega \subseteq \mathbb R^n$ measurable abd $f: \Omega \rightarrow \mathbb R$ be measurable. Assume $f \geq 0$ on $\Omega$. Then there exists a sequence $\{f_n: \Omega \rightarrow \mathbb R\}_{n=1}^\infty$ s.t. all $f_n$ are simple, non-negative and increasing and $f_n \rightarrow f$ pointwise on $\Omega$. Tao gives the hint […]

Fubini's theorem on a product of locally compact spaces which do not have countable bases

Let $X$ be a locally compact Hausdorff space. Let $\mathcal B$ be the $\sigma$-algebra generated by the family of open subsets of $X$. A measure $\mu$ on $\mathcal B$ is called a (positive) Radon measure if it satisfies the following conditions. 1) $\mu(K) \lt \infty$ for every compact set $K$. 2) $\mu(U) = sup \{\mu(K) […]