Check that the approximating sequence \(\left(f_{n}ight)_{n \in \mathbb{N}}\) for \(u\) in Theorem 8.8 consists of \(\sigma(u)\)-measurable functions.

Data from theorem 8.8

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(sombrero lemma) Let (X, A) be a measurable space. Every pos- itive A/B(R)-measurable function u: X [0,0] is the pointwise limit of an increasing sequence of simple functions fn = E(A), fn>0, i.e. u(x) = sup fn(x) = lim fn(x), f