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.
Suppose is a set, is a σ-algebra of subsets of , is a countably additive measure, and is a finitely additive map. We say that is absolutely continuous with respect to if for any there is such that for any such that . We say that is truly continuous with respect to if for any we can find and such that is finite and whenever and .
If we assume to be countably additive, then the definition of absolutely continuity simplifies as follows: is absolutely continuous with respect to if and only if for any we have whenever .
is truly continuous with respect to if and only if it is countable additive, absolutely continuous with respect to , and for any such that we can find such that , is finite, and .
Suppose is a set, is a σ-algebra of subsets of , is a countably additive measure, and is a function. Then there is a -integrable function such that for all if and only if is finitely additive and truly continuous with respect to .
In the context of the above theorem, if is σ-finite?, then there is a -integrable function such that for all if and only if is countably additive and absolutely continuous with respect to . If is finite, it suffices to require that is finitely additive.
The following example shows that the condition of being truly continuous is necessary in the non-σ-finite case. Consider an uncountable set , a σ-algebra of subsets of that are countable or have countable complement, the counting measure , and the measure that vanishes on countable sets and takes value 1 on all uncountable subsets. Then is absolutely continuous with respect to because only vanishes on the empty set. However, is not truly continuous with respect to because takes finite values only on finite sets, but for any such a set we have , which yields a contradiction if .
The measure space in the last example is quite pathological: it is not localizable?. A localizable? example can be constructed if and only if real-valued-measurable cardinals? exist. In this case, take to be a real-valued-measurable cardinal, to be the counting measure, and a probability measure that vanishes on all countable subsets of . Then is absolutely continuous with respect to because only the empty subset of has -measure 0. However, is not truly continuous with respect to because takes finite values only on finite sets, but for any such a set we have , which yields a contradiction if .
Truly continuous countably (or finitely) additive measures on a measure space form a module over the complex *-algebra of measurable maps modulo equality almost everywhere. The Radon–Nikodym theorem then says that this module is a free module of rank 1. Furthermore, generators of this module can be identified with truly continuous measures such that is absolutely continuous with respect to .
Suppose is a set, is a σ-algebra of subsets of , is a countably additive measure, and is a countably additive function. Then there is a unique decomposition
where is truly continuous with respect to , is absolutely continuous with respect to and vanishes on every measurable set of finite -measure, and is singular with respect to , meaning there is a set such that and for all such that . In particular, setting yields a unique decomposition of as a sum of a singular and absolutely continuous measure.
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 .
If is absolutely continuous with respect to and both and are -finite, then a Radon–Nikodym derivative of with respect to exists.
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
or equivalently
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.
Now let be Lebesgue measure on the real line and let be an upper semicontinuous function on the real line, so that defines a Borel measure generated by
Then is absolutely continuous if and only if is absolutely continuous, in which case the derivative exists almost everywhere and is a Radon–Nikodym derivative of . That is,
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
and
where is the identity function on the real line.
A comprehensive treatment can be found in Chapter 23 of
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:
Last revised on June 13, 2020 at 03:19:58. See the history of this page for a list of all contributions to it.