---

title: Measure & Probability

date: '2026-07-20 18:18:00'

tags:

- Statistics

cover: https://img.yulecna.top/posts/yn-background.jpg

description: ''

---
**Definition chain**
1. set $\mathcal{X}$ 
2. a nice enough subset of $2^{\mathcal{X}}$ ($\sigma$-field $\mathcal{F}$) 
3. **measure** $\mu$ over measurable space $(\mathcal{X}, \mathcal{F})$ 
	1. Non-negative
	2. countable additivity
	3. Empty set maps to zero
4. Push-forward measure $f$ over some measure space $(\mathcal{X}, \mathcal{F}, \mu)$
5. **Integral/Lebesgue Integral**: $\int f(x)d\mu(x)=\int fd\mu$
	1. Indicator function: $\int 1_A d\mu = \mu(A)$
	2. Simple function
	3. Approximating from below: 
	   $$
	   \int fd\mu=\lim_{n=1}^{\infty}\int f_n d\mu
	   $$
	   For example, assume: 
	   $$
	   f_n(x)=2^{-n}\cdot\lfloor2^nf(x)\rfloor=\sum_{k=1}^{\infty}k2^{-n}1_{A_{n,k}}(x)
	   $$ 
	   where 
	   $$
	   A_{n,k}=\{x: f(x)\in[k2^{-n}, (k+1)2^{-n})\}
	   $$
	4. Write $f(x) = f^+(x)-f^-(x)$
	5. e.g. **Expectation for Gaussian Distribution**: define $P_Z(A)=\int 1_A(X)\phi(x)dx=\mathbb{E}[1_A(Z)]$, by extending indicator to a general function, get $\int f d P_Z = \mathbb{E}[f(Z)]$
6. Densities: 
	1. Radon–Nikodym derivation ($P\ll\mu$)
	2. **Definition (the way to handle distributions): define distributions by giving their density wrt some known measure**
$$
\boxed{  
	P\ll\mu  
	\Longrightarrow  
	\exists,p=\frac{dP}{d\mu}  
	\text{ such that }  
	P(A)=\int_Ap\,d\mu
	\Longrightarrow
	\int fdP = \int f\,p\,d\mu
}
$$
7. Probability spaces and random variables: 
	1. It's too explicit to define a probability measure as the joint distribution of $(X, Y, Z, W)$. 
	2. abstract outcome $w$, abstract outcome space $\Omega$ $\longrightarrow$ event ($\omega$), random variable ($f(\omega)$).
	3.  think of $\mathbb{P}$ as a measure on $\Omega$ and $\mathbb{E}$ as an integral with respect to $\mathbb{P}$: 
	   $$
	   \begin{gather} \mathbb{P}(X<Y) = P(\{\omega\in\Omega: X(\omega)<Y(\omega)\}) \\ \mathbb{E}(X+Y)=\int X(\omega)+Y(\omega) \, d\mathbb{P}(\omega) \end{gather}
	   $$
8. Conditional probability: 
	1. Original definition: $\mathbb{P}(A|B)=\mathbb{P}(A\cap B)/\mathbb{P}(B)$
	2. Define expectation $\mathbb{E}(Y|X)$ as a random probability $g(X)$. 
	3. $\mathbb{P}(Y\in A|X) = \mathbb{E}[1_A(Y)|X)]$.