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

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How should I compute the integral of the Cantor-Lebesgue function ∫_0^1〖F(x)〗 dm?
 
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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.
 
I posted this question on math-stackexchange but apparently I asked something stupid and I was downvoted. I still don't have an answer to my question so I hope someone in here can help me or at least explain me why I am asking something stupid. I started studying Complex Analysis and came upon the following theorem which is a direct consequence of the Cauchy-Goursat theorem: Let ##f:D\to\mathbb{C}## be an anlytic function over a simply connected region ##D##. If ##a## and ##z## are part of...

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