The Lebesgue measure is a special measure defined on subsets of Euclidean spaces and is the standard way of measuring the length, area, and volume of these subsets. In this chapter, we define the Lebesgure measure and present some relevant theorems.

**Definition 4.1** (Lebesgue-Stieltjes Measure). A Lebesgue-Stieltjes measure on **R** = (−∞, ∞) is a measure μ on such that μ() < ∞ for each bounded interval ⊂ **R**.

**Definition 4.2** (Distribution Function). A distribution function on **R** is a map *F* : **R** → **R** that satisfies the following conditions:

(a) *F* is increasing; that is, *a* < *b* implies *F*(*a*) ≤ *F*(*b*).

(b) *F* is right-continuous:

where *x* → *x*^{+}_{0} means that *x* > *x*_{0} and *x* converges to *x*_{0}.

**Definition 4.3** (Lebesgue Measure). The Lebesgue-Stieltjes measure μ on **R** defined by

is called the *Lebesgue measure* on (see Theorem 4.1). The completion

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