Sentence examples for measurable selections of from inspiring English sources

Since the inf and the sup operators preserve measurability, we see that the functions l F and r F are measurable selections of F. □.

The next lemma shows that l F and r F are measurable selections of F when the latter is measurable.

We present some regularity properties for the set of distributions induced by the measurable selections of a correspondence over a Loeb space, which include closedness, convexity, compactness, purification, and semicontinuity.

Then the point functions l F and r F are measurable selections of F. Proof Choose a sequence of measurable selections { f n } n = 1 ∞ of F such that F = ⋃ n ≥ 1 f n ¯.

Then there exists a sequence ({ f_{n}}) of measurable selections of φ such that, for every x, varphi(x)=overline{bigl{ f_{n}(x mid n inmathbb{N}bigr} }= overline {bigcup_{n inmathbb{N}} f_{n}(x)}, where the bar denotes the closure in X. (cf., e.g., Lemma 7.1 in [36]).

In fact, since the multivalued composition (F_{q}(t)) is, for every (q in Q), obviously measurable, according to Proposition 1, we can even write y(t) inint_{0}^{T} G_{1} t, s) overline{bigcup_{ninmathbb{N}} f_{n, q} (s)}, mathrm{d}s, where ({f_{n, q}(t) subset F_{q}(t) }_{n inmathbb {N}}) is a sequence of measurable selections of (F_{q}) and the integral is understood in the sense of Aumann.

A measurable function is called a measurable selection of if for every Denote by (2.4).

Let be two measurable multivalued operators, be a constant and be a measurable selection of.

A measurable mapping is called a measurable selection of the operator if for each.

A mapping is said to be a measurable selection of a measurable mapping, if is measurable and a.e.

Then there exists a measurable selection of such that, for any, (21).