Given two measures on the same measurable space, their Radon–Nikodym derivative is essentially their ratio , although this is traditionally written because of analogies with differentiation. This ratio or derivative is a measurable function which is defined up to equality almost everywhere with respect to the divisor . It only exists iff is absolutely continuous with respect to .
Integration on a general measure space can be seen as the process of multiplying a measure by a function to get a measure. Then the Radon–Nikodym derivative is the reverse of this: dividing two measures to get a function.
Let be a measurable space (so consists of a set and a -algebra ), and let and be measures on , valued in the real numbers (and possibly taking infinite values) or in the complex numbers (and taking only finite values). Let be a measurable function (with real or complex values) on .
The function is a Radon–Nikodym derivative of with respect to if, given any measurable subset of , the -measure of equals the integral of on with respect to :
(The latter two expressions in this equation are different notations for the same thing.)
These properties are basic to the concept; the notation is as in the definition above.
Let be a Radon–Nikodym derivative of with respect to , and let be a measurable function on . Then is a Radon–Nikodym derivative of with respect to if and only if and are equal almost everywhere with respect to .
If a Radon–Nikodym derivative of with respect to exists, then is absolutely continuous with respect to .
For fairly elementary proofs, see Bartels (2003).
(This last theorem is not as general as it could be.)
Note the repetition of ‘with respect to ’ in various guises; let us fix (assumed to be -finite) and take everything with respect to it. Then it is convenient to treat all measurable functions up to equality almost everywhere; and given any absolutely continuous (also assumed to be -finite), we speak of the Radon–Nikodym derivative of .
See also the discussion of notation at measure space.
Using the simplest notation for integrals, the definition of Radon–Nikodym derivative reads
In other words, the measure is the product of the function and the measure :
and so is the ratio of to :
So this is the simplest notation for the Radon–Nikodym derivative.
However, this notation for integrals is uncommon; one is more likely to see
which leads to
for the Radon–Nikodym derivative. But none of these ‘’s are really necessary.
We can also use a fuller notation with a dummy variable as the object of the symbol ‘’:
this leads to
which does not give a symbol for directly. If instead of one unwisely writes , then this gives the previous notation for the Radon–Nikodym derivative.
The presence of ‘’ on the right-hand side inspires people to put it on the left-hand side as well; but this is spurious, since we really want to write
where is the identity function on the real line.
Some fairly elementary proofs prepared for a substitute lecture in John Baez's introductory measure theory course are here:
The strategy there is based on: