I was trying to solve the following simple integration involving indicator function $I_{(a,b]}$ in a journal article. Here are the equations (in LaTeX notation): $ f(u) = \int_{0}^{1} (I_{(0,s]}(u) - s)\; ds\tag{1} $ $ g(u,v) = \int_{0}^{1} (I_{(0,s]}(u) - s)(I_{(0,s]}(v) - s)\; ds\tag{2} $ where $0 < u, v < 1$. I was thinking that the integration will be simply just $ f(u) = \int_{0}^{1} (1 - s)\; ds\tag{1} $ $ g(u,v) = \int_{0}^{1} (1 - s)(1 - s)\; ds \tag{2} $ But, I'm not so sure about this. The constraint on both $u$ and $v$ confused me. Any pointer to this solution?
Thanks
Wayan