# touchstone idea – Measurability of significant supremum of duty of two variables Answer

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touchstone idea – Measurability of significant supremum of duty of two variables

For the memoir, here’s a unostentatious respond primarily based on Tonelli/Fubini’s theorem.

For capricious $$c in mathbb R$$, the clique $$M_c := {(x,y) in X occasions X mid f(x,y) > c}$$ is measurable, thus the indicator duty $$chi_{M_c}$$ is measurable (all the pieces w.r.t. the product touchstone). Tonelli/Fubini’s theorem tells us that $$h colon X to [0,infty]$$,
$$h_c(x) := int_X chi_{M_c}(x,y) , mathrm d y qquad forall x in X,$$
is measurable.
Finally,
$${ x in X mid g(x) > c } = bigl{ x in X bigm| mu({y in X mid f(x,y) > c }) > 0 bigr} = { x in X mid h_c(x) > 0 }$$
is measurable for all $$c in mathbb R$$.
Thus, $$g$$ is measurable.

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