useful. On a standard probability space a conditional expectation may be treated as the integral over the conditional measure (regular conditional probabilities, see also disintegration of measure). Given two standard probability spaces, every homomorphism of their measure algebras is induced by some measure preserving map. Every probability measure on a standard measurable space leads to a standard probability space. The product of a sequence (finite or not) of standard probability spaces is a standard probability space. All non-atomic standard probability spaces are mutually isomorphic mod 0; one of them is the interval (0,1) with the Lebesgue measure. These spaces are less geometric. In particular, the idea of dimension, applicable (in one form or another) to all other spaces, does not apply to measurable, measure and probability spaces. Non-commutative geometry The theoretical study of calculus, known as mathematical analysis, led in the early 20th century to the consideration of linear spaces of real-valued or complex-valued functions. The earliest examples of these were function spaces, each one adapted to its own class of problems. These examples shared many common features, and these features were soon abstracted into Hilbert spaces, Banach spaces, and more general topological vector spaces. These were a powerful toolkit for the solution of a wide range of mathematical problems. The most detailed information was carried by a class of spaces called Banach algebras. These are Banach spaces together with a continuous multiplication operation. An important early example was the Banach algebra of essentially bounded measurable functions on a measure space X. This set of functions is a Banach space under pointwise addition and scalar multiplication. With the operation of pointwise multiplication, it becomes a special type of Banach space, one now called a commutative von Neumann algebra. Pointwise multiplication determines a representation of this algebra on the Hilbert space of square integrable functions on X. An early observation of John von Neumann was that this correspondence also worked in reverse: Given some mild technical hypotheses, a commutative von Neumann algebra together with a representation on a Hilbert space determines a measure space, and these two constructions (of a von Neumann algebra plus a representation and of a measure space) are mutually inverse. Von Neumann then proposed that non-commutative von Neumann algebras should have geometric meaning, just as commutative von Neumann algebras do. Together with Francis Murray, he produced a classification of von Neumann algebras. The direct integral construction shows