MHB Compute the integral of the Cantor-Lebesgue function ∫_0^1〖F(x)〗 dm .

[Jack3]

How should I compute the integral of the Cantor-Lebesgue function ∫_0^1〖F(x)〗 dm?

 

[Jhoneine]

The integral of the Cantor-Lebesgue function is 0. This is due to the fact that the Cantor-Lebesgue function is a singular function and thus has a zero Lebesgue measure. To compute this integral, one can use the definition of the Lebesgue integral: ∫_0^1〖F(x)〗 dm = lim_{n→∞} ∑_i=1^n F(x_i)Δm_iwhere x_i are points in the interval [0,1] and Δm_i is the Lebesgue measure of the interval between them. Since the function is singular, all of the intervals between the points will have a measure of 0, so the integral will be 0.