# 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, f*_{1}, *f*_{2}, ··· 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 ...**

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