inverse image of infinity of measurable functions is measurable

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Theorem

If $f : E \to R$ is a measurable function, then $f^{- 1} ({\infty})$ and $f^{- 1} ({- \infty})$ are measurable

Proof

We can write

f^{- 1} ({\infty}) = ⋂_{n = 1}^{\infty} f^{- 1} ((n, \infty])

And because each set in the countable intersection is measurable, so too must the intersection. A similar argument follows for $f^{- 1} ({- \infty})$ . And using the equivalent intervals of measurability for measurable functions, we get the desired result

\begin{matrix} ◼ \end{matrix}

Combined with inverse image of measurable functions of all borel sets are measurable, this means that {1||the inverse image of any Borel set, including ${\pm \infty}$ } is always {2||measurable} for {3||measurable functions}.

File	Last Modified
Functional Analysis Lecture 9	2025-07-14
inverse image of measurable functions of all borel sets are measurable	2025-07-14

inverse image of infinity of measurable functions is measurable

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