integral of a non-negative function

Once we can integrate simple staircase functions, we approximate any non-negative measurable function from below by such staircases and ask how big we can make the integral. The answer — the supremum over all simple functions that stay underneath — is declared to be the integral. It is the natural fill-it-up-from-below construction: pack the region under the graph with the best simple shapes you can.

Formally, for a measurable f greater than or equal to 0, the integral of f dμ is the supremum of the integrals of all simple functions s with 0 less than or equal to s less than or equal to f. This supremum always exists in the extended sense, so the value lies in [0, infinity]; it may be plus infinity. No finiteness is required at this stage — non-negative functions always have a well-defined (possibly infinite) integral.

This definition is monotone (bigger function, bigger or equal integral) and, crucially, it is exactly the setup that makes the monotone convergence theorem work: an increasing sequence of non-negative functions can be integrated termwise in the limit. Finiteness is the separate question of integrability, addressed by requiring the integral of the absolute value to be finite.