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 \(\mathbb{R}\):

\[P((a, b]) := F(b) - F(a)\]