David Fremlin is writing the definitive tome on measure theory. He completed four volumes and is currently working on a fifth. You can find it all at the book’s web site.
For a more succinct account of measure theory, see James Hirschorn’s survey article.
A pseudodifferential operator is a generalization of a partial differential operator to fractional orders. Pseudodifferential operators allow you in some cases to invert a differential operator. For elliptic boundary value problems, they provide the easiest means to show that the surprising fact that the solution is smoother in the interior than on the boundary.
Here are some introductions to the subject: