Lebesgue Integral

Подпись: n sup m = 1 Подпись: X(Im ),

Подпись: b j f (x )dx
Подпись: mf f(x) x єІт

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.

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