chapter  15
Egorov type theorems
ByGilbert W. Bassett Jr., Roger Koenker
Pages 14

One of the earliest important results in real analysis and Lebesgue measure theory was obtained by Egorov [62], who discovered close relationships between the uniform convergence and the convergence almost everywhere of a sequence of real-valued Lebesgue measurable functions. This classical result (widely known now as Egorov’s theorem) has numerous consequences and applications in mathematical analysis and measure theory. For example, it suffices to mention that another classical result in real analysis – Luzin’s theorem on the structure of Lebesgue measurable functions – can easily be deduced by starting with the Egorov theorem.