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Which of the following statements cannot be always true ?

  • A.Let \(X : \Omega \to \Omega^{*}\) and if \(\tau_{j}\) be the \(\sigma\)-field of subsets of \(\Omega^{*}\), then \(X^{-1}(\tau_{j})\) is also \(\sigma\)-field
  • B.\(\sigma\{X^{-1}(\tau)\} = X^{-1}\{\sigma(\tau)\}\) where, \(\tau\) denote the class of subsets of \(\Omega^{*}\) and \(X : \Omega \to \Omega^{*}\)
  • C.If \(B \cap C = \phi\), then \(X(B) \cap X(C) = \phi\) where \(X(A) = \{X(w) \mid w \in A\}\)
  • D.\(X \ge 0\) and \(E(X) = 0\), then X = 0 a.s.

Answer: C

The statement that need not hold is B ∩ C = φ implies X(B) ∩ X(C) = φ.

Direct images do not preserve disjointness. If X is constant, say \(X(w) = 1\) for all w, then for disjoint non-empty B and C, \(X(B) = X(C) = \{1\}\), which intersect.

The others are standard results. Inverse images preserve complements and countable unions, so \(X^{-1}(\tau_{j})\) is a \(\sigma\)-field; the \(\sigma\)-field generated by \(X^{-1}(\tau)\) equals \(X^{-1}(\sigma(\tau))\); and if \(X \ge 0\) with \(E(X) = 0\), then \(P(X \gt 1/n) \le nE(X) = 0\) for every n, so X = 0 almost surely.

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