ABSTRACT
Let ( X , X , μ ) $ (X, {\mathcal{X}},\text{ }\mu ) $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781351056823/5c92c9a4-ddd5-4d42-82bc-8e5903ddaaca/content/inline-math12_1.tif"/> be a measure space. In this book we have treated four different types of convergence of a sequence of {f j } functions to a limit function f:
pointwise convergence: For each ɛ > 0 and each x ∊ X there is a number J > 0 such that, if j > J, then |f j (x) - f(x)| < ɛ.
uniform convergence: For each ɛ > 0 there is a number J > 0 such that, if j > J and x ∊ X, then |f j (x) - f(x)| < ɛ.
convergence almost everywhere: There exists a set E ⊆ X with μ(E) = 0 so that, for every ɛ > 0, and each x ∈ X \ E $ x \in X\backslash E $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781351056823/5c92c9a4-ddd5-4d42-82bc-8e5903ddaaca/content/inline-math12_2.tif"/> , there is a number J > 0 such that, for j > J, we have |f j (x) - f(x)| < ɛ.
convergence in L p , 1 ≤ p < ∞: For each ɛ > 0 there is a number J > 0 so that, if j > J, then
