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 […]

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 […]

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} […]

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 […]

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 […]

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 […]

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 […]

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 […]

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 […]

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) […]

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