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Key Terms

Conditioning & Functors

\(\sigma\)-algebra of a Random Variable

Information. What information about the sample space, \(\Omega\), does a random variable, \(X\), convey.

For any element of the Borel \(\sigma\)-algebra we can tell whether that element occured.

\[X(\omega) \in B, \quad \textrm{or} \quad X(\omega) \notin B\]

Therefore, we know

\[ \omega \in X^{-1}(B), \quad \textrm{or} \quad \omega \notin X^{-1}(B)\]

We refer to this set of events as the \(\sigma\)-algebra generated by \(X\). Importantly, it is the set of events that we can tell whether or not they occured if we have \(X\).

Conditioning on Random Variables

  • As we have explained else where, conditioning on an event can be understood as mapping a probility measure into another probability measure.
\[\mathcal{C}:: \mathcal{M} \to \mathcal{F}_+ \to \mathcal{M}\]
  • Events are isomphic to indicator random variables, where the \(\sigma\)-algebra of the indicator random variable for the event \(A\) is the following
\[\{\emptyset, \Omega, A, A^c \}\]
  • Conditioning on a random variable can then be understood as "lifting" the above function!
\[\tilde{C} :: \mathcal{M} \to (\Omega \to \mathcal{R}) \to \mathcal{F}_{\sigma(X)} \to \mathcal{M}\]

The question is, what do we do with this?!