chapter  4
18 Pages

Lebesgue Measurable Functions

Wewill use the concept of Lebesguemeasure to introduce a rich class of functions and amethod of integrating these functions. In this chapter, we describe the class of functions.

Let E be a measurable set in Rn. Let f be a real-valued (in the usual extended sense) function defined on E, that is, −∞ ≤ f (x) ≤ +∞, x ∈ E. Then, f is called a Lebesgue measurable function on E, or simply a measurable function, if for every finite a, the set

{x ∈ E : f (x) > a}

is a measurable subset of Rn. In what follows, we shall often use the abbreviation {f > a} for {x ∈ E : f (x)> a}. Note that the definition of measurability of a function on a set E presupposes that E is measurable. Since

E = { f = −∞} ∪ ( ∞

k=1 { f > −k}

)

,

the measurability of E implies that of { f = − ∞} if we assume that f is measurable.