Discussion
Let \(\Omega = \{1, 2, 3, \ldots, 10\}\), A = {1, 2} and \(\mathbf{F} = \{\phi, A, A^{C}, \Omega\}\), consider the following functions :
\[ X(w) = 100\ \forall\, w \in \Omega \]
\[ Y(w) = \begin{cases} 100 & \text{if } w \in A \\ 400 & \text{if } w \notin A \end{cases} \]
\[ Z(w) = w\ \forall\, w \in \Omega \]
Then which of the following is true ?
Answer: A
Only X and Y are F-measurable.
A function is F-measurable when the set where it takes each value (more generally each inverse image of a Borel set) lies in F = {φ, A, AC, Ω}.
X is constant, so its inverse images are \(\phi\) or \(\Omega\): measurable. Y takes the value 100 on A and 400 on \(A^{C}\), so its inverse images are \(\phi\), A, \(A^{C}\) or \(\Omega\): measurable.
Z(w) = w takes ten different values. The set {Z = 3} = {3} is not in F, so Z is not measurable. That rules out 'all three', 'only X' and 'none'.
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