Let $\alpha:[0,1] \to R$ be the Cantor function. Evaluate $$\int_{0}^{1}xd\alpha $$and $$\int_{0}^{1}x^2d\alpha.$$
I know that the Cantor function is continuous and monotone increasing, how can I evaluate the integral above using the properties of $\alpha$? Can someone help me solve this question?
For the first integral we have by parts$$\int_{0}^{1}xd\alpha\left(x\right)=\left.x\alpha\left(x\right)\right|_{0}^{1}-\int_{0}^{1}\alpha\left(x\right)dx=1-\int_{0}^{1}\alpha\left(x\right)dx $$ and since $\int_{0}^{1}\alpha\left(x\right)dx $ is the area of the Cantor function on $\left[0,1\right] $ we get $$\int_{0}^{1}xd\alpha\left(x\right)=\frac{1}{2}. $$ For the second integral see here.