# Definition:Filtered Probability Space

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## Definition

### Discrete Time

Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.

Let $\sequence {\mathcal F_n}_{n \mathop \in \N}$ be a discrete-time filtration of $\Sigma$.

We say that $\struct {\Omega, \Sigma, \sequence {\mathcal F_n}_{n \mathop \in \N}, \Pr}$ is a **filtered probability space**.

### Continuous Time

Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.

Let $\sequence {\mathcal F_t}_{t \ge 0}$ be a continuous-time filtration of $\Sigma$.

We say that $\struct {\Omega, \Sigma, \sequence {\mathcal F_t}_{t \ge 0}, \Pr}$ is a **filtered probability space**.

## Also known as

A filtered probability space may be known as simply a **filtered space**.