Sentence examples for be a measurable map from inspiring English sources

Lemma 3.8 (Dzedzej and Gelman [36]) Let F: [0, α] → Pc, cp(ℝ n ) be a measurable map such that the Lebesgue measure μ of the set {t: dim F t) < 1} is zero.

Let ( X, d ) be a complete separable metric space and let F : Ω → C L ( X ) be a measurable map.

Let X be a nonempty set, S be a measurable space which consists of some metrics on the X, ((Omega, P)) be a complete probabilistic measure space and (f : Omegarightarrow S) be a measurable mapping.

Let E be a nonempty set, S be a measurable space which consists of some metrics on E, ((Omega, P)) be a complete probabilistic measure space and (f : Omegarightarrow S) be a measurable mapping.

Let E be a linear space, S be a measurable space which consists of some norms on E, ((Omega, P)) be a complete probabilistic measure space and (f : Omegarightarrow S) be a measurable mapping.

Example Let X be a nonempty set, S be a measurable space which consist of some metrics on the X, ( Ω, P ) be a complete probabilistic measure space and f : Ω → S be a measurable mapping.

Let (x:Omegato H) be a measurable mapping and (theta:Omegato 0,+infty)) be a measurable function, then for all (x t)in H) and for each (tinOmega), (R_{theta(t)}^{phi} t,x t))) is a gap function for the RGVIP (2.1).

Let g : X → R be a measurable mapping satisfying that g and f are ⪯-comonotonic for all f ∈ R. Let us see that given x, y ∈ X with x ⪯ y ( x ≠ y ) there exists a mapping l ∈ R with l ( x ) < l ( y ), which implies that g ( x ) ≤ g ( y ), and so g is ⪯-preserving, that is, g ∈ R. If the above result is false, there exist x, y ∈ X with x ≠ y such that x ⪯ y and l ( x ) = l ( y ) for all l ∈ R.

Moreover, by the separability of the space L 2 ( Ω ; R ), conditions (i), (ii), (iii), and by the Pettis measurability theorem (see Theorem 2.4), we obtain that g is a measurable map, and so we have obtained for every z ∈ L 2 ( Ω ; R ) the existence of a measurable selection of F ( ⋅, z ).

The symbol Ω will denote a bounded open subset of R n and we shall always consider the case n ≥ 2. With O ⊂ R n being a measurable subset with positive measure, and with g : O → R n, n ≥ 1, being a measurable map, we shall denote by Open image in new window its integral average; here | O | denotes the Lebesgue measure of O.

Clearly, each ν n is a measurable map into L 2 ( D ).