Integral of cantor function

[robertdeniro]

Homework Statement

consider the ternary cantor set C, and the asscoiated cantor function f, and the associated Lebesgue-Stieltjes measure u.

what is the integral of f over all of R with respect to u?

Homework Equations

The Attempt at a Solution

i know that under the lebesgue measure, the integral of the cantor function is 1/2 using a symmetry argument.

but under this measure, u, we can break up the integral into 3 parts: (-infty, 0), (0, 1), and (1, infty).

since the cantor function is constant on (-infty, 0) and (1, infty), it follows that the integral is 0 on those intervals. I am stuck here, help would be appreciated

 

[mighty2000]

.

Hello,

You are correct in your reasoning so far. Since the cantor function is constant on the intervals (-∞, 0) and (1, ∞), the integral over those intervals will be 0.

For the interval (0, 1), we can use the fact that the cantor function is a staircase function, meaning it is a piecewise constant function with countably many discontinuities. This allows us to break up the integral into smaller intervals where the function is constant.

In this case, we can break up the interval (0, 1) into the intervals (1/3, 2/3) and (2/3, 1). The cantor function is constant on each of these intervals, with the value of 1/2 on (1/3, 2/3) and the value of 1/4 on (2/3, 1).

Using the definition of the Lebesgue-Stieltjes integral, we can calculate the integral over (0, 1) as follows:

∫f d u = ∫(1/2) d u + ∫(1/4) d u = (1/2)u(2/3) - (1/2)u(1/3) + (1/4)u(1) - (1/4)u(2/3) = (1/4)u(1) - (1/4)u(1/3) - (1/4)u(2/3)

Since u(1) = 1 and u(1/3) = u(2/3) = 1/2, the integral becomes:

∫f d u = (1/4) - (1/8) - (1/8) = 1/2

Therefore, the integral of the cantor function over all of R with respect to the Lebesgue-Stieltjes measure is 1/2.

I hope this helps! Let me know if you have any further questions.