The topology of close approximation

AbstractThree topologies on function spaces are considered; in increasing order of fineness they are: convergence in measure, uniform and “close approximation”. For simplicity and definiteness, attention is concentrated on the space of Lebesgue measurable real-valued functions on the real line; considerable generalizations are possible. The Borel classification of some families of functions, and the behavior of “most” functions (that is, of a residual set), are studied; the compact families of functions in the “close approximation” topology are characterized; and the effect of identifying functions modulo null sets is investigated