Let f be measurable with f^-1((a,∞))inA for all a∈ℝ. What does this imply about f^-1([a,∞))?

That it is not necessarily in A

Exercise · ZW-MEASURABLE-FUNCTION-009 · ★★☆ Intermediate

Let \(f\) be measurable with \(f^{-1}((a,\infty))\in\mathcal{A}\) for all \(a\in\mathbb{R}\). What does this imply about \(f^{-1}([a,\infty))\)?

Choose one option.

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D. That it also belongs to \(\mathcal{A}\)

Explanation

Since \([a,\infty)\) is the intersection of the sets \((a-1/n,\infty)\), its preimage is a countable intersection of measurable sets and so lies in the σ-algebra. It is neither empty nor equal to the open preimage.

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Review the concept: Measurable function

A function f: X → ℝ is measurable with respect to a σ-algebra \(\mathcal{A}\) on X if the preimage \(f^{-1}(B)\) belongs to \(\mathcal{A}\) for every Borel set B of ℝ.

\[f^{-1}(B)\in\mathcal{A}\ \forall B\in\mathcal{B}(\mathbb{R})\]

Let \(f\) be measurable with \(f^{-1}((a,\infty))\in\mathcal{A}\) for all \(a\in\mathbb{R}\). What does this imply about \(f^{-1}([a,\infty))\)?

Explanation

Review the concept: Measurable function

More practice problems: measurable function