Probability Measures
Parent: Probability
Definition
A measure is a function from a sigmaalgebra to the reals \(\mu: F
\rightarrow \mathbb{R}\), satisfying two properties:

\[\forall A \in F, \mu(A) \geq \mu(\varnothing) = 0\]
 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. \(\mu(\Sigma) = 1\).
 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 rightcontinuous, nondecreasing function \(F\) tending to 0 at \(\infty\)
and to 1 at \(\infty\), there exists a unique measure defined on Borel sets of \(\mathbb{R}\):
\[P((a, b]) := F(b)  F(a)\]