When is measurable, from (2.6), we obtain that is measurable.
where a : Ω × R → S N ( R ), b : Ω × R × R N → R N, f : Ω × R → R are Carathéodory functions (that is, they are measurable in x ∈ Ω and continuous in the other variables), g ∈ H 1 , and S N ( R ) denotes the space of N × N sized symmetric matrices.
where ) are Carathéodory functions; that is, (resp., ) is measurable in (resp., in for each and continuous in for a.e. (resp.,. More general, the function can be a nonlocal function, for example, or. Let and be Banach spaces, we introduce some notations as follows: (i),.,.
We say that ψ is a Carathéodory function if it is measurable in t for each x in (mathbb{R}^{d}) and continuous in x for almost everywhere (a.e., for short) given (tin I).
(A1) Each satisfies Carathéodory conditions, that is, is measurable in for all and continuous in for a.e.. Furthermore, a constant and a function exist so that (3.1).
Each satisfies Carathéodory conditions, that is, is measurable in for all and continuous in for a.e.. Furthermore, a constant and a function exist so that (4.35).
That positive change is measurable in other ways as well.
The maps and are Carathéodory functions, which means that they are measurable in the first argument and continuous in the second one.
f, g ∈ Car ( R × R, R ) are the L 2 -Carathéodory functions, which means that they are measurable in the first variable and continuous in the second variable, and for every 0 < r < s there exists h r, s ∈ L 2 [ 0, T ] such that | g ( t, x ( t ) ) | ≤ h r, s for all x ∈ [ r, s ] and a.e.
We will assume that the following conditions hold: (a1 For all, the elements, of the matrix function are measurable in the square, the elements, of the vector function are measurable in the interval, (2.2). (a2) is a measurable scalar function satisfying,.
That's a significant roar that ought to be measurable in future.
