Measure Theoretic Probability
$(\Omega, \mathcal{A}, \mathbb{P})$ is a probability space if $(\Omega, \mathcal{A})$ is a measurable space and $\mathbb{P}$ is a measure on $\mathcal{A}$ such that $\mathbb{P}(\Omega) =1$.Our goal is to understand what a probability space is, which in turn consists of a measurable space and a measure. So in the following we will define them.
We will first define what an algebra on a set $\Omega$ is and then we will define what a $\sigma-$algebra is, $\Omega$ and a $\sigma-$algebra on $\Omega$ define a measurable space. A measure is a function from a $\sigma-$algebra to the reals. A measure space is a triplet consisting of a set $\Omega$, a sigma algebra $A$ on it and a measure $\mu$ on the sigma algebra.
Algebra: A collection of subsets of a set $\Omega$ is called an algebra $A$ if :
$\sigma-$Algebra: A collection of subsets of a set $\Omega$ is called sigma algebra $A$ if :
Hence, $\sigma-$Algebra additionally requires closure not only under pairwise union but also under countably infinite union.
Note: $\{A_i\}_{i\in I}$ are some sigma algebras on $X$, then $\bigcap_{i\in I} A_i$ is also a sigma algebra. Note that $\bigcap_{i\in I} A_i$ is the smallest sigma algebra with all the properties in $\{A_i\}_{i\in I}$
Example of algebras that is not a $\sigma$-algebra:
$\sigma(M)$ or sigma algebra generated by $M$, where $M \subseteq P(X)$ is some set is the smallest sigma algebra containing $M$. i.e. : $$\sigma(M):= \bigcap_{A \subseteq M }A$$ where $A$ are all possible sigma algebras on $X$.
Topological Space [via Open sets]: A topology on a set $X$ may be defined as a collection $\tau$ of subsets of $X$, called open sets and satisfying the following axioms:
Borel-Sigma Algebra: The Borel-sigma algebra denoted by $B$, of the Topological space $(X,\tau)$ is the sigma algebra generated by the family $\tau$ of open sets. It's elements are called Borel Sets.
Lemma: Let $C = \{(a,b): a < b\}$. Then $\sigma(C) = B_R$ is the borel field generated by the family of all open intervals in $C$.
Measurable Space: The pair of a set $\Omega$ and a $\sigma-$ algebra on it.
Measure $\mu$ is a function defined on a measurable space $(\Omega, A)$ from it's $\sigma-$ algebra to reals. Such that:Measure Space is a triplet $(\Omega,A,\mu )$
Probability Measure $\mathbb{P}$ is a measure such that $\mu(\Omega) = 1$
Probability Space is a measure space $(\Omega,A,\mathbb{P} )$
Continuity An equivalent formulation of countable additivity is continuity. $\mathbb{P}$ is a finitely additive and continuous if for any decreasing sequence $B_n \in A$ such that $B_n \subseteq B_{n+1}$, $$B = \bigcap_{n \geq 1 } B_n \rightarrow \lim_{n \rightarrow \infty} \mathbb{P}(B_n) = \mathbb{P}(B)$$
Continuity $\leftrightarrow$ Countable additivity.
Lebesgue Measure: There is a unique measure $\lambda$ on $(\mathbb{R},B_{\mathbb{R}})$ that satisifes:
$$\lambda([a,b]) = b-a$$
for every finite interval $[a,b]$, $-\infty < a \leq b < \infty$. This is called the Lebesgue measure.
If we restrict $\lambda$ to the measure space $([0,1], B_{[0,1]})$, then $\lambda$ is a probability measure.
Zero Measure: A $\mu$-measurable set $E$ is said to have $\mu$-measure zero if $\mu(E) = 0$
Almost everywhere:
A particular property is said to hold "almost everywhere" if the set of points for which the property does not hold has a measure zero.Example: "A function vanishes almost everywhere", "$f=g$ almost everywhere"
Kolmogorov's Axioms define a list of axioms for a probability measure. Let $P:E\rightarrow [0,1]$ be a probability measure and $E$ be some sigma algebra generated by X.