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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 $$\mathbb{R}$$:

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