Convergence in Law of Measurable Processes with Applications to the Prediction Process

We study convergence in law of measurable processes with a general state space and a parameter set. The space of measurable functions are first investigated and we examine properties of probability measure on the space. A necessary and sufficient condition for convergence in law of measurable processes is obtained. These general results are applied to the prediction process, and we show that convergence of the prediction processes implies that of given processes. We also find a simple condition for convergence of the prediction processes when given processes are Markovian.