Integral

Let be a Measure Space
Given a Simple Function:

we define the integral:

For any nonnegative Measurable Function
we define its integral as:

Finally, for any Measurable Function with ,
we say that is integrable and we define the integral:

where and .

Monotone Convergence
Fatou’s Lemma
Dominated Convergence
Differentiation under the integral
Fubini’s Theorem