Exercise 1.3

Question

Prove that if $f$ is a real function on a measurable space $X$ such that $\{x : f(x) \ge r\}$ is measurable for every rational $r$, then $f$ is measurable.

Amit Awasthi · Accepted Answer

\begin{proof}
      By Theorem $1.12(c)$, it suffices to show that $f^{-1}((\alpha,\infty))$
      is measurable for every $\alpha \in \mathbf{R}$. So let $r_n$
      be a sequence of rationals such that $r_n 	o \alpha$ as $n 	o \infty$.
      Then, $(\alpha,\infty) = \bigcup_n [r_n,\infty)$, and the inverse image of
      each $[r_n,\infty)$ is measurable, hence the inverse image of
      $(\alpha,\infty)$ is also.
\end{proof}

Exercise 1.3

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Exercise 1.3

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