Probability Notes but more Formally

Once the philosophy has been stripped away, probability theory is simply the study of an object, a probability distribution that assigns values to sets, and the transformations of that object.

Probability is a positive conserved quantity that we can distribute across a space. This is that more formal notion of spreading ‘cream cheese’.

A measure space is a mathematical object that is defined by a triple: $(X,A,\mu )$ where $X$ is a set, $A$ is a $\sigma$-algebra on the set and $\mu$ is a measure. A measure is a particular kind of function that maps from the $A$ space to the real number line.

Measure theory aims to abstract the notion of ‘size’ ¹.

The measure function assigns a size to each element in A. It is the function $\mu :A\to \Re$

Intuitively each element of the $\sigma$-algebra is a part of a larger whole. So that if we add up each individual element we get some ‘whole’ (countable additivity). It’s a particular set of subsets of X.

We define a measurable space as $(X,A)$ the initial set and $\sigma$-algebra, the subsets that can be measured. The $\sigma$-algebra set is often the power set of X.

Measurable function (probability)

A function P is a probability measure for the probability space $(\Omega ,F)$ if it satisfies:

• $0\le P(A)\le 1$ for $A\in F$.
• $P(\varnothing )=0,P(\Omega )=1$
• $A_1,...A_n$ is a disjoint sequence of F sets, then $P({\cap }_{k=1}^{\infty }{A}_k)={\sum }_{k=1}^{\infty }P(A_k)$.

Basics

Here we start measuring the ‘size’ of uncertainty.

Define $\Omega$ as the unit interval (0, 1]. The length of the unit interval $I$ is $|I|=b-a$, $A={\cup }_{i=1}^n{I}_i$

The set $\Omega$ can represent all future possible worlds.

Provided A is disjoint and finite and lies in $\Omega$ then we assign a measure of probability

$P(A)={\sum }_{i=1}^n(b_i-a_i)$

A is a set of subsets that is a $\sigma$-algebra.

Functions/Transforms

A probability distribution is a mapping of the form $\pi :X\to [0,1]$ for each atomic element in X.

Expectation

A distribution as defined allows us to summarise the distribution function with numbers.

Reduction of functions of the form $f:X\to R$ to a single number.

If $C$ is the space with all functions of the form $f:X\to R$ then expectation is a map $f$

Definitions

A space is usually denoted by its set and structure $(X,s)$. An example here might be the ordering of the set with some ordering rule (like in decision theory).

The indicator, or indicator function, of a set A is the function on $\Omega$ that assumes the value 1 on A and 0 on $A^C$ it is denoted $I_A$.

Footnotes

1. https://mbernste.github.io/posts/measure_theory_1/ ↩