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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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