The class (mathcal{C}(Jtimesmathbb{R}times mathbb{R},mathbb{R})) is called the Carathéodory class of functions on (Jtimesmathbb{R}timesmathbb{R,}) which are Lebesgue integrable when bounded by a Lebesgue integrable function on J.
The class Car ( J × R, R ) is called the Carathéodory class of functions on J × R which are Lebesgue integrable when bounded by a Lebesgue integrable function on J.
The class ({mathcal{C}}(J timesmathbb{R},mathbb{R})) is called the Carathéodory class of functions on (J timesmathbb{R} ) which are Lebesgue integrable when bounded by a Lebesgue integrable function on J. the map (t mapsto g t,x)) is measurable for each (x in mathbb{R} ), and. the map (x mapsto g t,x)) is continuous for each (t in J).
The first equality follows because the function inside the integral tends to zero at every point and is upper bounded by the following integrable function, c ′ ε 0 y + 1 1 ∂f ∂Θ Θ, Π ( i, y ) ≤ 1 + c ′′ ε 0 y + 1 2 1 ∂f ∂Θ Θ, Π ( i, y ) > 1.
They also considered the cases of synchronous functions as well as the functions bounded by integrable functions.
Lemma 3.5 shows that (b_{n}^{v}(C_{n}(x D_{n}(x -1)e^{-x -1mbda(x)) is bounded by integrable function independent of n.
Conversely, if g is Riemann integrable, it is bounded by Lemma 2.2.
The second term on the right hand side of (25) tends to zero by bounded convergence theorem and mean value theorem (as in Appendix 2), because Π Θ ≤1everywhere for all Θ and as ∇g is uniformly bounded in a neighborhood of Θ by an integrable function.
Given a real-analytic function b(x) defined on a neighborhood of the origin with b(0)="0, we consider local convolutions with kernels which are bounded by |b(x)|−a, where a>0 is the smallest number for which |b(x)|−a is not integrable on any neighborhood of the origin.
The scheduling signals, although bounded and square integrable on a finite time interval, are not constrained to be linear, but rather are permitted to be a nonlinear function of external parameters or time.
