Probability Measures

Why

A probability event function is a measure on the measurable space consisting of the set of outcomes and the set of events.

Definition

A probability measure is a finite measure on a measurable space which assigns the value one to the base set. A finite measure can always be scaled to a probability measure, so these measures are standard examples of finite measures.

A probability space is a measure space whose measure is a probability measure. The outcomes of a probability space are the elements of the base set. The set of outcomes is the base set. The events are the elements of the sigma algebra. The measure in a probability space corresponds to the event probability function.

Notation

Let $(X, \mathcal{A} )$ be a measurable space.¹ We denote the sigma-algebra by $\mathcal{A} $, as usual. We denote a probability measure by $\mathbfsf{P} $, a mnemonic for “probability,” and intended to remind of the event probabilty function.²

Many authors associate an event $A \in \mathcal{A} $ with a function $\pi : X \to \set{0, 1}$ so that $A = \Set*{x \in X}{\pi (x) = 1}$. In this context, it is common to write $\mu [\pi (x)]$ for $\mu (A)$.

1. Often, other authors will denote the set of outcomes (here denoted by $X$) by $\Omega $, an apparent mnemonic for “outcomes”. ↩︎
2. Often, other authors will use notation including $P$, $Pr$, $\text{Prob}$, and $\text{pr}$. ↩︎