Avony

Definition chain

set $\mathcal{X}$
a nice enough subset of $2^{\mathcal{X}}$ ( $\sigma$ -field $\mathcal{F}$ )
measure $\mu$ $μ$ over measurable space $(\mathcal{X}, \mathcal{F})$ $(X, F)$
1. Non-negative
2. countable additivity
3. Empty set maps to zero
Push-forward measure $f$ over some measure space $(\mathcal{X}, \mathcal{F}, \mu)$
Integral/Lebesgue Integral: $\int f(x)d\mu(x)=\int fd\mu$ $\int f (x) d μ (x) = \int f d μ$
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 $\int f d P_Z = \mathbb{E}[f(Z)]$
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 }

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 $\mathbb{P}$ : $\begin{gather} \mathbb{P}(X<Y) = P(\{\omega\in\Omega: X(\omega)<Y(\omega)\}) \\ \mathbb{E}(X+Y)=\int X(\omega)+Y(\omega) \, d\mathbb{P}(\omega) \end{gather}$
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. $\mathbb{P}(Y\in A|X) = \mathbb{E}[1_A(Y)|X)]$ .