Measure, Probability, and Mathematical Finance: A Problem-Oriented Approach

CHAPTER 9

CONVERGENCE

In measure theory, there exist several different notions of convergence of measurable functions, for example, convergence almost everywhere, convergence in measure, convergence in L^p, to just name a few. In this chapter, we present definitions of some common notions of convergence.

9.1 Basic Concepts and Facts

Definition 9.1 (Convergence in Measure). Let f, f₁, f₂, ··· be Borel measurable functions on a measure space (S, ∑, μ). The sequence {f_n}_n≥1 is said to converge to f in measure, written as , if and only if for every > 0, μ{s : |f_n(s) − f(s)| ≥ } → 0 as n → ∞.

In particular, if μ is a probability measure, then convergence in measure is also referred to as convergence in probability.

Definition 9.2 (Convergence in L^p). Let f be a Borel measurable function on a measure space (S, ∑, μ) and {f_n}_n≥1 a sequence of of functions in L^p(S, ∑, μ). The sequence {f_n}_n≥1 is said to converge to f in L^p, written as , if and only if ||f_n − f||_p → 0 as n → ∞, where .

Definition ...