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-Löf 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).

Ergodic theory is concerned with dynamical systems -- collections of points together with a rule governing how the system changes over time. Much of the theory is concerned with the long term behavior of typical points-- how points behave over time, ignoring anomalous behavior from a small number of exceptional points. Computability theory has a family of precise notions of randomness: a point is "algorithmically random'' if no computable test can demonstrate that it is not random. These notions capture something essential about the informal notion of randomness: algorithmically random points are precisely the ones that have typical orbits in computable dynamical systems. For computable dynamical systems with or without assumptions of ergodicity, the measure 0 set of exceptional points for various theorems (such as Poincaré's Recurrence Theorem or the pointwise ergodic theorem) are precisely the Schnorr or Martin-Löf random points identified in algorithmic randomness.

We discuss the different contexts in which relativization occurs in randomness and the effect that the relativization chosen has on the results we can obtain. We study several characterizations of the K-trivials in terms of concepts ranging from cuppability to density, and we consider a uniform relativization for randomness that gives us more natural results for computable randomness, Schnorr randomness, and Kurtz randomness than the classical relativization does (the relativization for Martin-Löf randomness is unaffected by this change). We then evaluate the relativizations we have considered and suggest some avenues for further work.