The idea of Lebesgue integral is the following: we give to a simple non-negative function $\sum_{j=1}^Na_j\chi_{S_j}$, where $a_j\geq 0$ and $S_j>0$ the value $\sum_{j=1}^Na_j\mu(S_j)$. Then we define the integral of a measurable non-negative function as $$\int_X f(x)d\mu(x):=\sup\{\int_X g(x)d\mu(x), 0\leq g\leq f,g \mbox{ simple}\}.$$ For a measurable function, write $f=\max\(f,0\)-\max\(-f,0\)$ to give a value to $\int_X f(x)d\mu(x)$.
The major interest is that we can integrate functions with are defined in an arbitrary set, provided we have putted a $\sigma$-algebra and a measure on it.
