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A problem from Rudin's Real and Complex Analysis

Measure and Integration Theory
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r9m
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A problem from Rudin's Real and Complex Analysis

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Post by r9m » Wed May 24, 2017 8:11 pm

Problem: Let, $f$ be a real-valued Lebesgue measurable function on $\mathbb{R}^k$, prove that there exists Borel functions $g$ and $h$ such that $\mu_k(\{x \in \mathbb{R}^k: g(x) \neq h(x)\}) = 0$ and $g(x) \le f(x) \le h(x)$ a.e. $[\mu_k]$.

(where, $\mu_k$ is the Lebesgue measure on $\mathbb{R}^k$)

Note: This is problem 14 from pg-59 of Rudin's Real and Complex Analysis book. The problem as stated in book is incorrect. It requires us to find Borel functions $g$ and $h$ such that $\mu_k(\{x \in \mathbb{R}^k: g(x) \neq h(x)\}) = 0$ and $g(x) \le f(x) \le h(x)$ for all $x \in \mathbb{R}^k$. It's interesting to construct a counter-example to the above proposition/disprove it.
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