On Baire property of spaces of compact-valued measurable functions

A topological space X is Baire if the Baire Category Theorem holds for X, i.e., the intersection of any sequence of open dense subsets of X is dense in X. One of the interesting problems in the theory of functional spaces is the characterization of the Baire property of a functional space through the topological property of the support of functions. In the paper this problem is solved for the space $M(X, K)$ of all measurable compact-valued (K-valued) functions defined on a measurable space (X,Σ) with the topology of pointwise convergence. It is proved that $M(X, K)$ is Baire for any metrizable compact space K.