Lebesgue integral of non-negative functions is defined as:
$$\int f(x)dx=\sup_g\int g(x)dx$$
where the sup is taken over all measurable functions $g$ such that $0\leq g\leq f$ and $g$ bounded, $m(supp(g))<\infty$.
My Questions: Suppose $g(x)\le M, m(supp(g))\le N$, then $\int g(x)dx\le M\cdot N$ should hold. But why would it be possible that $\int f(x)dx=\infty$