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 $(X,\mathrm{\Sigma })$ with the topology of pointwise convergence. It is proved that $M(X,K)$ is Baire for any metrizable compact space $K$.