Sentence examples for measurable functions of from inspiring English sources

The entries (a_{ij} y)) of the matrix A are measurable functions of y.

In this section, we assume that the coefficients (A^{alpha beta }) are measurable functions of (x in mathbb {R}^d).

Throughout this section, we consider the operator (mathcal {L}_0), i.e., the coefficients (A^{alpha beta }) are measurable functions of (x_1) only.

Let L p ( R ) denote the class of measurable functions of ϕ on ℝ such that the integral ∫ R | ϕ ( x ) | p d t is finite.

Proposition 1 essentially tells that if the graph is explored by a Markov chain-based sampling algorithm and the samples are taken as any measurable functions of the underlying Markov chain, satisfying Condition (1) then Condition D ″ holds.

end{aligned}Our first results concern a priori (L_q -estimates of the Stokes system defined in (mathbb {R}^d) or (mathbb {R}^d_+), when the coefficients (A^{aL_q -estimatesre merely measurable functiofs of only (x_1).

If T is a sublinear map from L p 0 + L p 1 to the space of measurable functions on Y which is of weak-types (p0,q0) and (p1,q1), then T is of type (p, q).

Example 2.8 For 0 < p < 1, let L p (0,1) be the space of equivalence classes of measurable functions on [0,1], with ∥ f ∥ = ∫ 0 1 ∣ f ( t ) ∣ p d t < ∞.

For, let be the space of measurable functions on, such that (2.5).

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

Suppose that is a measurable function of -degree in and for any it satisfies and (3.5).