Sentence examples for Borel measurable function on from inspiring English sources

for a nonnegative Borel measurable function on and a nonnegative measure on a Borel set.

Let k be a non-negative Borel measurable function on ({mathbf{R}}^{n}times{mathbf{R}}^{n}), and set k y,mu)=int_{E}k y,x), dmu(x quad mbox{and}quad k( mu,x =int_{E}k y,x), dmu y) for a non-negative measure μ on a Borel set (Esubset{mathbf{R}}^{n}).

Let f be a Borel measurable function of the complex plane to itself.

Furthermore, (Q (x theta)), where (x=(x_{1},ldots,x_{n},x_{n})), is a Borel measurable function of x for any fixed (thetainTheta) and a continuous function of θ for any fixed (xinmathbf{R}^{n}).

From now on: F : R + × R + → R + is a Borel measurable function, j : R + → R is a non-increasing function.

Let x be an F-observable on a fuzzy measurable space ((X, M)) and (f: R^{1} to R^{1}) be a Borel measurable function.

Let be a real-valued Borel measurable function, symmetric in its arguments.

Let f : R → R be a Borel measurable function taking values in an interval I ⊂ R such that fφ is integrable, and let q be a convex function on I such that ( q ∘ f ) φ is integrable.

Each w j ( t ) is scalar standard Brownian motion defined on a complete probability space ( Ω, F, P ) with a natural filtration { F t } t ⩾ 0. The noise perturbation σ i j : R × R → R is a Borel measurable function.

Then Y is σ ( X ) -measurable if and only if Y = g ( X ) for some Borel measurable function g: R d → R d. Lemma A.2 (Doob's martingale convergence theorem).

Since the function (f: R^{1} to R^{1}) is a Borel measurable function, (f^{ - 1}(E) in B(R^{1})), and hence (a in R x)).