ag.algebraic geometry - Lie bracket on the unshifted tangent complex?

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

Hello expensive customer to our community We will proffer you an answer to this query touchstone idea – Measurability of significant supremum of duty of two variables ,and the respond will breathe typical by documented info sources, We welcome you and proffer you fresh questions and solutions, Many customer are questioning concerning the respond to this query.

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.

we are going to proffer you the answer to touchstone idea – Measurability of significant supremum of duty of two variables query through our community which brings all of the solutions from a number of dependable sources.

Add comment