Abstract
We introduce the notion of a measurable function needed in the construction of the integral, study the properties of measurable functions and different types of convergence for sequences of measurable functions. We prove the important theorem of Egorov, which reduces pointwise convergence to uniform convergence by deleting an appropriate set of arbitrarily small measure. We discuss the question of approximation of measurable functions by continuous ones.KeywordsMeasurable FunctionUniform ConvergenceSimple FunctionZero MeasureInverse ImageThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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