Theorems of Lebesgue integration theory which is useful for calculating integral

Question

It is well known that fundamental theorem of calculus and residue theorem are very useful for calculating integral.

However I haven't heard such things in Lebesgue integration theory. Are there any such theorems? The ones only for particular measure spaces are ok.

Edit:"such" means "useful for calculating integral value directly". Some special concrete situations are also ok.

Answer 1

The Tonelli and Fubini theorems are two of the most important results from measure theory to allow multi-dimensional integrals to be calculated over sets that aren't suitably elementary.

Furthermore, the Lesbegue Dominated Convergence Theorem shows up all the time to calculate limits and derivatives of intgrals.