Almost Discrete convergence

A sequence of functions {fn} is said to be (D)-convergent to f on a set X if for every x 2 X there exists a positive integer n0 = n0(x) such that fn(x) = f(x) for all n n0, and it is said to be (D a.e.)-convergent to f on X if {fn} is (D)-convergent to f on X rM where M is of measure zero. We extend this definition when fn and f are measurable. The sequence {fn} is said to be (AD)-convergent to f on X if for every " > 0 there is n" such that μ({x 2 X: fk(x) 6= f(x) for some k n}) < " for n > n" and it is said to be (DM)-convergent to f on X if for every " > 0 there is n" such that μ({x 2 X: fn(x) 6= f(x)}) < " for n > n". We conduct a comparative study of (D a.e.)-convergence, (AD)-convergence and (DM)-convergence and their relations with “almost everywhere convergence”, “almost uniform convergence”, and “convergence in measure”. They also show with the help of (DM)-convergence that the space M(X) of all measurable functions on X is complete with respect to the metric d defined on M(X) by d(f, g) = μ({x 2 X: f(x) 6= g(x)}), assuming, as usual, f and g to be identical if they are equal almost everywhere.