UNIFORM CONVERGENCE THEOREMS MOTIVATED BY DINI’S THEOREM FOR A SEQUENCE OF FUNCTIONS

Being motivated by Dini’s Theorem, the uniform convergence of a sequence of functions is characterized by that of another decreasing sequence of functions. This result allows us to describe the uniform convergence of a sequence of measurable functions on a closed interval in terms of that of a decreasing sequence of Lebesgue integrable functions. For a sequence of continuous functions on a compact set, its uniform convergence is further characterized by the pointwise convergence of another decreasing sequence of continuous functions. It also turns out that the uniform convergence of a sequence of Riemann integrable functions on a closed interval can be determined by that of another decreasing sequence of Riemann integrable functions. As applications, characterizations for a uniformly convergent series of functions have been presented and one fixed point theorem has been established.