Click measure_theory_terence_tao_book1.pdf link to view the file.

We have discussed the idea of the size of sets, both finite and infinite. We have also examined the ideas of lengths, areas and volumes. The underlying concepts admit of elegant generalizations and abstractions. part of these are covered under what is called the theory of measure. The attached piece, `Introduction to Measure Theory', by Terence Tao, is a very succinct and very accessible introduction to the basic questions addressed by this theory, and the core of the answers.

I might point out that the theory of measure is not to be confused with the (representational) theory of measurement. More on this in the lectures. For now, over to Terence Tao and his charmingly written notes. Incidentally, Tao is a Field Medalist, and he was on a visit recently to TIFR. Look him up on the net.