The theory of regular conditional probability is generalized by replacing a probability measure by a (perhaps non- σ \sigma -finite) outer measure and a resulting measure is obtained on the product space. A Fubini-like theorem is obtained for the integrable functions of this measure and a condition is given for this measure to impart the topological properties of being inner regular and almost Lindelöf to the product space when the component spaces also have these topological properties. Thus some theorems for the Morse-Bledsoe product measure [1] are generalized by methods very similar to those used in their paper on product measures [1].