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What is the filtered probability space?
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A filtered probability space, or stochastic basis, $\left(\Omega, \mathcal{F},\left(\mathcal{F}_{t}\right)_{t \in T}, \mathbb{P}\right)$ consists of a probability space $(\Omega, \mathcal{F}, \mathbb{P})$ and a filtration $\left(\mathcal{F}_{t}\right)_{t \in T}$ 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 $\mathcal{F}_{t}$

Given any filtered probability space, it can always be enlarged by passing to the completion of the probability space, adding zero probability sets to $\mathcal{F}_{t}$, and by replacing $\mathcal{F}_{t}$ by $\mathcal{F}_{t+} .$ This will then satisfy the usual conditions. In fact, for many types of processes defined on a complete probability space, their natural filtration will already be right-continuous and the usual conditions met. However, the process of completing the probability space depends on the specific probability measure $\mathbb{P}$ and in many situations, such as the study of Markov processes, it is necessary to study many different measures on the same space. A much weaker condition which can be used is that the $\sigma$-algebras $\mathcal{F}_{t}$ are universally complete, which is still strong enough to apply much of the 'heavy machinery' of stochastic processes, such as the Doob-Meyer decomposition, section theorems, etc.
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