Sentence examples for real-valued measurable function from inspiring English sources

Next, as an application, we consider some geometric properties of φ-means of a real-valued measurable function f on Ω.

We introduce a mean of a real-valued measurable function f on a probability space induced by a strictly monotone function φ.

Throughout the paper, we denote by, I and f a probability space, an interval of ℝ and a real-valued measurable function on Ω with f ∈ I for almost all ω ∈ Ω, respectively.

(1.5) If f is a real-valued measurable function defined on (Omega_{2}), the general Hardy-Knopp type operator (A_{k}) is defined by A_{k}f(x):=frac{1}{K x)}int_{Omega_{2}}k x,y f y),dmu_{2} y), quad xinOmega_{1}.

For each real-valued measurable function f on Ω, let C s m *, f 2 ( I ) be the set of all C2-functions φ in C sm,f (I) with no stationary points, that is, φ'(t) ≠ 0 for all t ∈ I. Lemma 8. Let φ, ψ ∈ C s m *, f 2 ( I ).

Let be an arbitrary fuzzy measure on and be two real-valued measurable functions such that and.

Denote by L 0 = L 0 [ 0, α ) the set of all m-equivalence classes of real-valued measurable functions defined on [ 0, α ) with m being the Lebesgue measure on ℝ and α = 1 or α = ∞.

We are interested in means of real-valued measurable functions induced by strictly monotone functions.

We denote by (mathfrak{M}(mathbb {R}^{n})) the set of all extended real-valued measurable functions on (mathbb {R}^{n}).

We denote by M ( R n ) the set of all extended real-valued measurable functions on R n and by M + ( 0, ∞ ) the set of all non-negative measurable functions on ( 0, ∞ ).

Let ({f_{n}}) be a sequence of real-valued measurable functions on a measurable set E. Suppose that the sequence converges pointwise to a function f and is dominated by some integrable function F in the sense that (vert f_{n}(x vert leq F x)) for all numbers n in the index set of the sequence and all points (x in E).