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Measure, Probability, and Mathematical Finance: A Problem-Oriented Approach by Hong Xie, Chaoqun Ma, Guojun Gan

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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 Lp, 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 {fn}n≥1 is said to converge to f in measure, written as , if and only if for every > 0, μ{s : |fn(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 Lp). Let f be a Borel measurable function on a measure space (S, ∑, μ) and {fn}n≥1 a sequence of of functions in Lp(S, ∑, μ). The sequence {fn}n≥1 is said to converge to f in Lp, written as , if and only if ||fnf||p → 0 as n → ∞, where .

Definition ...

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