Probability Space

Compact study note.

Summary

Probability space is the complete mathematical model for random experiment structure. It combines possible outcomes, measurable events, and probability measure.^[1]

Prerequisites

• Sample Space and Events
• Sigma-Algebra

Notation and Assumptions

Probability space notation:

It consists of sample space $\mathrm{\Omega }$ , event sigma-algebra $\mathcal{F}$ , and probability measure $P:\mathcal{F}\to [0,1]$ with $P(\mathrm{\Omega })=1$ and countable additivity over disjoint events.

Essential Result

All probability statements are statements about events in $\mathcal{F}$ . Random variables are measurable functions defined on this space.

Small Example

For a fair coin, $\mathrm{\Omega }=\{H,T\}$ , $\mathcal{F}=\mathcal{P}(\mathrm{\Omega })$ , and $P(\{H\})=P(\{T\})=1/2$ .

Common Mistakes

• Writing probabilities for sets outside $\mathcal{F}$ .
• Leaving $\mathcal{F}$ implicit in measure-theoretic arguments.

Connections

• Probability
• Conditional Probability
• Random Variable

References