Young Measures and Compactness in Measure Spaces

Convergence results for solutions of a first-order differential equation

“…The proof of this result can be found in [9, Theorem 3.84 and 3.85], (see also [7]). We recall also a result of compactness in measure proved in [9,Theorem 3.102], (see also [7,Theorem 3.12]). …”

“…If (u n ) n∈N is bounded in L 1 ([0, 1], E), then we can apply Theorem 3.70 and (2) of Remark 3.71 from [9]. It follows that we can extract the above subsequence so that u n − u 1 → η((u n )).…”

“…The theory begins with the works of L. C. Young (1937); the extensions to Polish and Suslin spaces were made by E. J. Balder (1984) and M. Valadier (1990). A general presentation of theory can be found in [9] (see also [3]). …”

“…The following theorem gives a justification for the denomination Jordan finite-tight set (see [9,Theorem 3.94] and [7,Theorem 3.4…”

On a fixed point theorem for the product of operators

“…It was observed that a compactness criterion for Lebesgue spaces requires some uniform integrability condition, as well as a condition of compactness in measure (see [22] or [26] for the discussion). Thus, the next goal of the paper is to indicate a big class of sets which are compact in measure.…”