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We take under consideration Young measures with densities. The notion ofdensity of a Young measure is introduced and illustrated with examples. It isproved that the density of a Young measure is weakly sequentially closed set.In the case when density of a Young measure is a singleton (up to the set ofnull measure), it is shown that the strong closedness (in rca(K)) of the set ofsuch measures, associated with Borel functions with values in the compact set K? Rl, is equivalent with the strong closedness (in L1(K)) of the set of theirdensities, provided the set K is convex. For an m-oscillating function thenotion of a total slope is proposed. It turns out, that if the total slopes ofthe elements of the sequence of oscillating functions form monotonic sequence,then the sequence of the respective (homogeneous) Young measures is weaklyconvergent in rca(K). The limit is a homogeneous Young measure with the densitybeing the weak L1 sequential limit of the densities of the underlying Youngmeasures.

Weak convergence of the sequences of homogeneous Young measures associated with a class of oscillating functions