Borel partitions of unity and lower Carathéodory multifunctions

We prove the existence of Carathéodory selections and representations of a closed convex valued, lower Carathéodory multifunction from a set A in ${\displaystyle A(ℰ\otimes ℬ\left(X\right))}$ into a separable Banach space Y, where ℰ is a sub-σ-field of the Borel σ-field ℬ(E) of a Polish space E, X is a Polish space and A is the Suslin operation. As applications we obtain random versions of results on extensions of continuous functions and fixed points of multifunctions. Such results are useful in the study of random differential equations and inclusions and in mathematical economics.   As a key tool we prove that if A is an analytic subset of E × X and if ${\displaystyle \{U_n:n\in w\}}$ is a sequence of Borel sets in A such that ${\displaystyle A={\cup }_n{U}_n}$ and the section ${\displaystyle U_n\left(e\right)}$ is open in A(e), e ∈ E, n ∈ w, then there exist Borel functions ${\displaystyle p_n:A\to [0,1]}$, n ∈ w, such that for every e ∈ E, ${\displaystyle \{p_n(e,·):n\in w\}}$ is a locally Lipschitz partition of unity subordinate to ${\displaystyle \{U_n\left(e\right):n\in w\}}$.