Standard Machinery

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Theorem

Let $\left({X, \Sigma, \mu}\right)$ be a measure space.

Let $\mathcal{L}^1_{\overline \R} \left({\mu}\right)$ be the space of $\mu$-integrable functions.

Let $P \left({f_1, \ldots, f_n}\right)$ be a proposition, where the variables $f_i$ denote $\mu$-measurable functions $f_i: X \to \overline{\R}$.

Let every occurrence of an $f_i$ be of the form:

$\displaystyle \int \Phi \left({f_i}\right) \, \mathrm d\mu$

for a suitable index set $I$ and multilinear mapping $\Phi: \mathcal{L}^1_{\overline \R} \left({\mu}\right)^I \to \mathcal{L}^1_{\overline \R} \left({\mu}\right)$.


Denote with $\chi \left({\Sigma}\right)$ the set of characteristic functions of elements of $\Sigma$, i.e.:

$\chi \left({\Sigma}\right) := \left\{{\chi_E: X \to \R: E \in \Sigma}\right\}$


Then the following are equivalent:

$(A): \quad \forall f_1, \ldots, f_n \in \chi \left({\Sigma}\right): P \left({f_1, \ldots, f_n}\right)$
$(B): \quad \forall f_1, \ldots, f_n \in \mathcal{L}^1_{\overline \R} \left({\mu}\right): P \left({f_1, \ldots, f_n}\right)$

that is, it suffices to verify $P$ holds for characteristic functions.

Proof