MathCS.org - Real Analysis: Example 7.4.4(a): Lebesgue Integral for Simple Functions

Example 7.4.4(a): Lebesgue Integral for Simple Functions

Find the Lebesgue integral of the constant function f(x) = c over the interval [a, b].

Context

The constant function f(x) = c can be written as a simple function

f(x) = c X_R(x)

Therefore, the (Lebesgue) integral of f over [a, b] is by definition:

_{[a, b]} f(x) dx = X_{[a, b]}(x) f(x) dx =
= c X_{[a, b]}(x) dx = c m([a, b]) = c (b - a)

which is the same answer as for the Riemann integral.

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