Is a function like $H(x) = \left\{ {\begin{array}{*{20}{c}} { 0}&{x < 0}\\ { + \infty }&{x = 0}\\ 1&{x > 0} \end{array}} \right.$ Lebesgue integrable on interval $(-1,1)$?
It looks to me that it is integrable because Lebesgue integral is defined as the least upper bound of the area under all simple functions below $f$. However my friend tells me it is not.
Thank you.