Discussion
Let \(\Omega = \{1, 2, ........\}\) the set of positive integers and \(\mathbf{F} = \{\phi, \{1\}, \{2\}, \{1, 2\},\) \(\{1\}^{c}, \{2\}^{c}, \{1, 2\}^{c}, \Omega\}\), where \(A^{c}\) denotes the complement of the set A. Let \(R = (-\infty, \infty)\) and \(\mathbf{B}\) the Borel \(\sigma\)-field of R.
Which of the following functions \(f\) from \((\Omega, \mathbf{F})\) to \((R, \mathbf{B})\) is measurable ?
Answer: D
The measurable function is \(f(1) = 2\), \(f(2) = 3\), \(f(k) = 10\) for \(k \ge 3\).
The \(\sigma\)-field F is generated by the three atoms {1}, {2} and \(\{3, 4, 5, \ldots\}\), and its sets are the unions of these atoms. A real function is F-measurable exactly when it is constant on each atom, in particular constant on \(\{3, 4, \ldots\}\).
Option D is constant there (value 10), so every preimage is a union of atoms. Option A takes different values 3, 4, ...; option B takes 1 at 3 and 0 beyond; option C alternates between 1 and 2 on the third atom. For instance, in option C the preimage of {1} is the set of odd numbers, which is not in F.
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