A filtered probability space, or stochastic basis, $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\in T},\mathbb{P})$ consists of a probability space $(\Omega,\mathcal{F},\mathbb{P})$ and a filtration $(\mathcal{F}_t)_{t\in T}$ contained in $\mathcal{F}$ Here, $T$ is the time index set, and is an ordered set -- usually a subset of the real numbers -- such that $\mathcal{F}_s\subseteq\mathcal{F}_t$ for all $s<t$ in $T$

Filtered probability spaces form the setting for defining and studying stochastic processes. A process $X_t$ with time index $t$ ranging over $T$ is said to be adapted if $X_t$ is an $\mathcal{F}_t$ measurable random variable for every $t$

When the index set $T$ is an interval of the real numbers (i.e., continuous-time), it is often convenient to impose further conditions. In this case, the filtered probability space is said to satisfy the usual conditions or usual hypotheses if the following conditions are met.

• The probability space $(\Omega,\mathcal{F},\mathbb{P})$ is complete.
• The $\sigma$ algebras $\mathcal{F}_t$ contain all the sets in $\mathcal{F}$ of zero probability.
• The filtration $\mathcal{F}_t$ is right-continuous. That is, for every non-maximal $t\in T$ the $\sigma$ algebra $\mathcal{F}_{t+}\equiv\bigcap_{s>t}\mathcal{F}_s$ is equal to $\mathcal{F}_t$