I am reading Royden's Real Analysis to learn about Lebesgue integration.

Royden first shows that a bounded function on a set of finite measure is Lebesgue integrable if and only if it is measurable. Then, he goes on to define the integral of a non-negative measurable function $f$ on a measurable set $E$ as:

$\int_E f = \sup\limits_{h \leq f} \int_E h$,

where $h$ is a bounded measruable function which vanishes outside a set of finite measure

What I am wondering about this definition is why we need a non-negative function $f$ to be measurable in the first place. Wouldn't the same definition be well defined even if $f$ is not measurable? In the case of the integral of a bounded function $f$ on a finite measure, all we assume is the boundedness of $f$, and the measurability of $f$ turns out to be the equivalent characterization of the Lebesgue integrability of $f$.


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