Probability Measures

Parent: Probability

Definition

A measure is a function from a sigma-algebra to the reals $\mu: F \rightarrow \mathbb{R}$, satisfying two properties:

1. $$\forall A \in F, \mu(A) \geq \mu(\varnothing) = 0$$
2. Countable additivity: the measure of the union of any countable disjoint subsets is equal to the sum of the measure of each subset i.e. $\mu(\bigcup_i A_i) = \sum_i \mu(A_i)$

A measure is on $F$ is called a probability measure on $F$ if the measure of the sample space is unity i.e. $\mu(\Sigma) = 1$.

Immediate Consequences

• Monotonicity: $A \subset B \Rightarrow \mu(A) \leq \mu(B)$
• Subadditivity: $A \subset \bigcup_{m=1}^{\infty} A_m \Rightarrow \mu(A) \leq \sum_{m=1}^{\infty} \mu(A_m)$
• Continuity from below: measures of sets $A_i$ in increasing sequence converge to the measure of limit $\bigcup_i A_i$
• Continuity from above: measures of sets $A_i$ in decreasing sequence converge to the measure of intersection $\bigcap A_i$

Constructing Measures

Constructing Measures on Reals

TODO: Clarify this (18.175 Lecture 1 Page 11)

Theorem: for each right-continuous, non-decreasing function $F$ tending to 0 at $-\infty$ and to 1 at $\infty$, there exists a unique measure defined on Borel sets of $\mathbb{R}$:

$$P((a, b]) := F(b) - F(a)$$