# A question on measurable functions.

The question is:

Show that a positive function $f$ on $E$ is $\varepsilon$-measurable if and only if it has the form $$f = \sum_{n=1}^{\infty} a_{n} I_{A_{n}}$$ for some sequence $\{ a_{n} \}_{n=1}^{\infty} \subseteq \overline{\mathbb{R}}_{+}$ and some sequence $\{ A_{n} \}_{n=1}^{\infty} \subseteq \varepsilon$.

I know that the absolute-value function from $\overline{\mathbb{R}}$ to $\overline{\mathbb{R}}_{+}$ is $\varepsilon$-measurable when $\varepsilon$ equals the Borel $\sigma$-algebra on $\overline{\mathbb{R}}$. But it doesn’t have the form $$f = \sum_{n=1}^{\infty} a_{n} I_{A_{n}},$$ since its image is uncountable. What am I getting wrong?

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Dear Yangzhe, I edited your question to make it clearer. I hope you don’t mind! :) – Haskell Curry Feb 28 '13 at 5:05
@HaskellCurry: Thank you very much for your help. – Yangzhe Lau Feb 28 '13 at 6:17

The range of $x\mapsto \sum_n a_n \, I_{A_n}(x)$ can be uncountable. Given $x\in E$, each choice of whether $x$ does or does not belong to $A_1$, whether $x$ does or does not belong to $A_2$, $\dots$, produces a different subset of the $a_n$s to sum. Since there are countably infinitely many choices, each of which can be made in two different ways, this permits the range of $x\mapsto \sum_n a_n\, I_{A_n}(x)$ to have cardinality up to $2^{\aleph_0}$. This is the cardinality of all of $\overline {\Bbb R}_+$, so there is no cardinality restriction on the range which prevents you from getting all measurable functions in the given way.