Probability Measure

A probability measure is a special kind of measure used in the foundation of probability theory. It assigns a value between 0 and 1 to certain subsets of a sample space, encapsulating the idea of the "probability" of events.

Definition:

Let $(\Omega, \mathcal{F})$ be a measurable space, where:

• $\Omega$ is the sample space, which represents the set of all possible outcomes.
• $\mathcal{F}$ is a σ-algebra on $\Omega$, representing the set of all events (subsets of $\Omega$ to which we can meaningfully assign a probability).

A probability measure $P$ on this measurable space is a function: $$ P: \mathcal{F} \to [0, 1] $$ that assigns to each event $A \in \mathcal{F}$ a number $P(A)$ in the interval $[0,1]$, satisfying the following axioms:

1. Non-negativity: For every event $A \in \mathcal{F}$, $P(A) \geq 0$.
2. Normalization: $P(\Omega) = 1$. This means the probability of the entire sample space (i.e., the probability that "something" happens) is 1.
3. Countable Additivity: For any countable collection of mutually exclusive (disjoint) events $A_1, A_2, \dots$, we have:

$$ P\left(\bigcup_{i=1}^{\infty} A_i\right) = \sum_{i=1}^{\infty} P(A_i) $$

The last axiom ensures that the probability of the union of disjoint events is the sum of their individual probabilities. This generalizes the idea that the probability of either of two non-overlapping events occurring is the sum of their probabilities.

Remarks:

• The axioms ensure that a probability measure generalizes our intuitive notions about probability. For example, if you think of $\Omega$ as the outcomes of rolling a fair six-sided die, $P({3})$ would represent the probability that the die shows a 3, which would be $1/6$ for a fair die.
• In the context of probability theory, measures that satisfy the above properties allow us to develop a consistent and rigorous mathematical framework for studying random phenomena.

In essence, while measures in general assign a "size" or "volume" to subsets of a space, probability measures assign a "likelihood" or "chance" to events in a probabilistic setting.