Sentence examples for measurable function for from inspiring English sources

Then closedness of F implies ξ is a mapping from Ω to F. Since F is a subset of a separable Banach space X, so, if T is a continuous random operator then by [17, Lemma 8.2.3], the map ω → T ω, f is a measurable function for any measurable function f from Ω to F. Thus {ξ t } is measurable. Hence ξ: Ω → F, being the limit of {ξ t }, is also measurable. Lemma 2.1. [15]).

where are essentially bounded measurable functions, are measurable functions for and are summable with respect to and measurable essentially bounded with respect to for.

Measurable functions, for example, can represent various attributes of the nodes such as income or frequency of messages in social networks.

Then, and satisfy one of the following items: (i) and is arbitrary,   (ii) and are bounded measurable functions,   (iii),,   (iv),,  . and is arbitrary, and are bounded measurable functions,,,,, for some, and a bounded measurable function.

For (1leq p<infty), let (L_{p}(mathbb{T}^{d})) denote the space of all measurable functions for which Vert fVert _{p}:= biggl(frac{1}{(2pi)^{d}} int_{mathbb{T}^{d}} biglvert f(x bigrvert ^{p},dx biggr)^{frac{1}{p}}< infty.

where q j ( t ) are essentially bounded measurable functions, τ j ( t ) are measurable functions for j = 1, …, m, and k i ( t, s ) are summable with respect to s and measurable essentially bounded with respect to t for i = 1, …, n.

Here, (fin L_{1,varphi }( mathbb {R} ^{2})) and (L_{1,varphi }( mathbb {R} ^{2})) are the space of all measurable functions for which (left| frac{f }{varphi }right|) is integrable provided (varphi : mathbb {R} ^{2}rightarrow mathbb {R} ^) is a weight function which is bounded on any bounded subset of (mathbb {R} ^{2}.) The paper is organized as follows: In Sect.

Recall that (L^p_w({mathbb {R}}^nu )), the weak (L^p) space is defined as the measurable functions for which (||f||_{p,w}^*) is finite where begin{aligned} |{x,|,|f(x)|>t}| le frac{(||f||_{p,w}^*)^p}{t^p} end{aligned} (7.40)(||f||_{p,w}^*) is defined to be the minimal constant so that (7.40) holds.

If (f : mathbb T rightarrow mathbb{R }) is Lebesgue measurable, and (Asubseteq mathbb T ) is Lebesgue measurable, we write Open image in new window (2.1)The set of all Lebesgue measurable functions for which (Vert fVert _{p,A}<infty ) is denoted by (L^p(A)), with the understanding that functions which are equal almost everywhere on (A) are considered equal as members of (L^p(A)).

(tin Omega) and (varphi(cdot, u)) is a Σ-measurable function for every (uinmathbb{R}) is called a generalized Orlicz function or a Musielak-Orlicz function.

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}).