Introduction to the Mathematical and Statistical Foundations of Econometrics
Lebesgue Integral
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The Lebesgue measure gives rise to a generalization of the Riemann integral. Recall that the Riemann integral of a nonnegative function f (x) over a finite interval (a, b] is defined as
where the Im’s are intervals forming a finite partition of (a, b] - that is, they are disjoint and their union is (a, b] : (a, b] = unm=1 Im — and A.(Im) is the length of Im; hence, X(Im) is the Lebesgue measure of Im, and the supremum is taken over all finite partitions of (a, b]. Mimicking the definition of Riemann integral, the Lebesgue integral of a nonnegative function f (x) over a Borel set A can be defined as
j f (x)dx = sup£ (xinf f (x)^ X(Bm),
where now the Bm’s are Borel sets forming a finite partition of A and the supre - mum is taken over all such partitions.
If the function f (x) is not nonnegative, we can always write it as the difference of two nonnegative functions: f (x) = f+ (x) — f—(x), where
f+(x) = max[0, f(x)], f— (x) = max[0, — f(x)].
Then the Lebesgue integral over a Borel set A is defined as
f f(x )dx = / f+ (x )dx — / f—(x )dx
A A A
provided that at least one of the right-hand integrals is finite.
However, we still need to impose a further condition on the function f in order for it to be Lebesgue integrable. A sufficient condition is that, for each Borel set B in R, the set {x : f (x) є B} is a Borel set itself. As we will see in the next chapter, this is the condition for Borel measurability off
Finally, note that if A is an interval and f (x) is Riemann integrable over A, then the Riemann and the Lebesgue integrals coincide.
