# Correct definition of the probability space of a $\mathcal{F}_{t}$-measurable random variable

Let the triple $$\left ( \Omega, \mathcal{F}, \mathbb{P} \right )$$ designates the probability space of a stochastic event and let us consider the filtration $$\left ( \mathcal{F}_{t} \right )_{t \in \mathcal{T}}$$, where $$\mathcal{T} = \left \lbrace 1,2,\dotsc, T \right \rbrace$$ and $$T \in \mathbb{N}$$.

Now, let the following sequence of $$\mathcal{F}_{t}$$-measurable random variables $$\left ( X_{t} \right )_{t \in \mathcal{T}}$$ be a stochastic process on the filtered probability space $$\left ( \Omega, \mathcal{F}, \left ( \mathcal{F} \right )_{t \in \mathcal{T}}, \mathbb{P} \right )$$.

I was wondering what of the following statements about a specific $$X_{t}$$ is formally correct:

• $$X_t$$ is a $$\mathcal{F}_{t}$$-measurable random variable on $$\left ( \Omega, \mathcal{F}, \mathbb{P} \right )$$
• $$X_t$$ is a $$\mathcal{F}_{t}$$-measurable random variable on $$\left ( \Omega, \mathcal{F}_{t}, \mathbb{P} \right )$$

## 1 Answer

We say that a variable $$X_t:\Omega \rightarrow \mathbb{R}$$ is $$\mathcal{F}_t$$-measurable if $$X_t^{-1}(A) \in \mathcal{F}_t$$ for all $$A \in \mathcal{B}(\mathbb{R})$$, where $$\mathcal{B}$$ is the Borel $$\sigma$$-algebra on the reals.

If by "on the filtered space" you mean adapted to the filtration, then yes, by definition $$X_t$$ is an $$\mathcal{F}_t$$-measurable random variable on $$(\Omega, \mathcal{F}_t,P)$$. The second statement is surely true.

Also, note that $$X_t^{-1}(A) \in \mathcal{F}_t \subset \mathcal{F}$$ for all $$t$$ and for all $$A\in \mathcal{B}(\mathbb{R})$$! Therefore, also the first statement must be correct.