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 $$\int_X f(x)d\mu(x):=\sup\left\lbrace \int_X g(x)\mathrm{d}\mu(x) \mid 0\leq g\leq f,\ g \text{ simple}\right\rbrace.$$ For a measurable function, write $f=\max(f,0)-\max(-f,0)$ to give a value to $\int_X f(x)\mathrm{d}\mu(x)$.

The major interest is that we can integrate functions with are defined in an arbitrary set, provided we have fixed a $\sigma$-algebra and a measure on it.

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