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'$\mathcal{F}_{t}$-measurable function'
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| Title of object: |
$\mathcal{F}_{t}$-measurable function |
| Canonical Name: |
MathcalF_tMeasurableFunction |
| Type: |
Definition |
| Created on: |
2006-09-22 18:25:17 |
| Modified on: |
2006-10-31 18:12:34 |
| Classification: |
msc:60A99 |
Revision comment (for changes between this and next version):
| Changes for correction #10768 ('\mathcal{F}'). |
Preamble:
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Content:
Let $(\Omega,\mathcal{F},P)$ be a probability space and $\{F_t\}_{t\geq 1}$ a filtration. A function $f\colon \Omega \to \mathbb{R}$ is called $\mathcal{F}_{t}$- measurable function if for each $t\in [0,\infty)$ fixed ,
$$f^{-1}(U)=\{\omega\in \Omega \colon f(\omega)\in U\} \in \mathcal{F}_{t}$$
for all open sets $U \in \mathbb{R}$, equivalently for all Borel sets $U\subset \mathbb{R}$. In this case we say that $f(\omega)$ is known by the time $t$. |
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