  We develop a theory of Lp spaces based on outer measures rather than
measures. This theory includes the classical Lp theory on measure spaces as
special case. It also covers parts of potential theory and Carleson embedding
theorems. The theory turns out to be an elegant language to describe aspects of
classical singular integral theory such as paraproduct estimates and T(1)
theorems, and it is particularly useful for generalizations of singular
integral theory in time-frequency analysis. We formulate and prove a
generalized Carleson embedding theorem and give a relatively short reduction of
basic estimates for the bilinear Hilbert transform to this new Carleson
embedding theorem.
