  This is a survey of constructive and computable measure theory with an
emphasis on the close connections with algorithmic randomness. We give a brief
history of constructive measure theory from Brouwer to the present, emphasizing
how Schnorr randomness is the randomness notion implicit in the work of
Brouwer, Bishop, Demuth, and others. We survey a number of recent results
showing that classical almost everywhere convergence theorems can be used to
characterize many of the common randomness notions including Schnorr
randomness, computable randomness, and Martin-L\"of randomness. Last, we go
into more detail about computable measure theory, showing how all the major
approaches are basically equivalent (even though the definitions can vary
greatly).
