By the Caratheodory-type selection theorem (see Appendix), there exists a function f i : U i → Y such that f i ( ω, x ˜ ) ∈ Φ i ( ω, x ˜ ) for all ( ω, x ˜ ) ∈ U i, for each x ˜ ∈ L X, f i ( ⋅, x ˜ ) is measurable on U i x ˜, for each ω ∈ Ω, f i is continuous on U i ω and, moreover f i is jointly measurable.
The function g i : Ω i → S i is said to be a selection of ( X i z i if g i ( ω i ) ∈ ( X i ( ω i ) ) z i for every ω i ∈ Ω i. D ( X i z i is the set { ( μ τ i − 1 ) g i − 1 : g i is a measurable selection of ( X i z i } and D X, z : = ∏ i ∈ I D ( X i z i. For each i ∈ I, we denote h g i = ( μ τ i − 1 ) g i − 1, where g i is a measurable selection of ( X i z i and h g = ( h g 1, h g 2, …, h g n ).
If for ω(x) = Σ I ω I (x dx I, ω I (x) is measurable on Ω, then Tω(x) = f x, ω(x)) is measurable on Ω. Proof.
For each i ∈ I, S i is a countable complete metric space and ( Ω i, Z i ) is a measurable space.
For each i ∈ I, the space of actions S i is a countable complete metric space and ( Ω i, Z i ) is a measurable space.
A function is said to satisfy Carathéodory conditions if (i) is measurable in for any ; (ii) is continuous in for almost all. . is measurable in for any ; is continuous in for almost all.
(i) is measurable for each (ii) is u.s.c. for almost all . is measurable for each. is u.s.c. for almost all.
where f, h are continuous function and T-periodic about t, h ( t, x ) ≤ 0, g : [ 0, T ] × ( 0, ∞ ) → R is an L 2 -Carathéodory function, i.e., it is 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, ω ] such that | f ( t, x ( t ) ) | ≤ h r, s for all x ∈ [ r, s ] and a.e.
