Probability Measures

The third component of a probability space is a probability measure \(P\) on the events in the event space \(\mathscr{F}\). A measure is just a mathematical function that assigns measurements to things being measured. For us, this means assigning probabilities to events. So \(P(A)\) is the probability of the event \(A\), where \(A \in \mathscr{F}\). That's really all we need to know here about how measures work.

But besides \(P\) being a measure, we also need \(P\) to behave in a way that comports with our expectations around how probabilities work. There are three foundational principles, or axioms, that \(P\) must satisfy to be a good probability measure. So let's state those.

Axioms of probability

Axiom 1: \(P(A) \geq 0\) for all \(A \in \mathscr{F}\). That is, a probability is defined for every event \(A \in \mathscr{F}\), and it's nonnegative.

Axiom 2: \(P(\Omega) = 1\). Intuitively, the probability for the whole sample space is 1, which means that when we run an experiment, at least one of the outcomes must occur.

Axiom 3: \(P(\bigcup_{i \in J}{A_i}) = \sum_{i \in J}{P(A_i)}\) for countable index set \(J\) and disjoint \(A_i\). This one is a little trickier to understand, but not too bad. For countably many \(A_i\), the probability of their union is just the sum of their individual probabilities. Which makes sense.

Now these are axioms, so that means that they're intended as an minimalistic/economical way to state the requirements on \(P\). But there some other factoids—factoids that follow more or less immediately from the axioms—that you'll need to know. We'll call these key properties for lack of a better term.

Key properties of probability