On this page:

• Introduction
• Motivation
• Signed Measures
• Structure Theorem: Radon-Nikodym
• References

Introduction

In measure theory, one comes across the notion of signed measures that defined as follows:

[definition] [deftitle]Definition %counter% (signed measures)[/deftitle]

A signed measure on the measurable space $(X, \mathcal{M})$ is a function

such that:

• $\nu(\emptyset) = 0$;
• $\nu$ assumes at most one of $-\infty$ or $+\infty$;
• if $(E_n)\subseteq \mathcal{M}$ is disjoint, then we have the following where the series converges absolutely provided the LHS is finite:

[/definition]

When I first learned about signed measures, I found the idea pretty confusing. I was given the definition verbatim out of Folland’s book, but never actually understood why we would care about such a thing in the first place. What’s an example problem that we would want to solve via signed measures? Sure, Folland says that we can classify the signed measures via Radon-Nikodym, but why should we care about that at all? I spent a bit of time pondering this recently and I came up with an answer that I think is somewhat satisfactory to me.

Motivation

Obviously, there are no shortages of applications of integrals. What actually sticks out to me so much in integration, in particular, is the Dirac delta function, a mysterious notational convenience to say the following:

The Dirac delta is ubiquitous throughout applications in that it is precisely the right tool to model point masses and instantaneous changes in a function and how that would affect integration. That being said, the above is notation. A natural mathematical question is to ask how it connects back to our usual intuition of integrals? What does it mean to differentiate these things (or, for that matter any kind of limiting operation)? Classically, we know there is no connection apriori. Additionally, we come across expressions like

where $A$ and $B$ are constants, in the wild (for example, we can think of these as random variables — one that is discrete and the other absolutely continuous) and a natural question is how to interpret such a thing in a formal setting. Under the much more general measure theoretic framework, we can make the relationship precise. This is a useful perspective because then we unlock the much more powerful tools of measure theory if we, say, want to talk about limiting-based operations on these kinds of sums.

Signed Measures

The naive way of thinking of measures is that measures give us the length/area/volume of a set interpreted in an appropriate sense. We are taught in calculus that the appropriate tool for these kinds of problems is the Riemann integral. Shortly after that, we are told that signed area is a thing (which has a variety of applications, negative rates of change or densities, for example). Thus, a natural generalization of a measure is to consider a more general “measure” of the form

where $E \subseteq \RR$. In general, $f$ need not be nonnegative so we call this more “general measure” a signed measure. But these are not the only kind of signed measures! For example, we know that there are signed measures of the form

in the wild as well! Following physicist intuition, the latter integral is obviously just

The first “integral” is pretty imprecise — we know it’s just a notational convenience at the naive level. But this makes it abundantly clear that this more general case, if we formalize it, includes strange-looking expressions like the above. By the Radon-Nikodym theorem, it turns out that these two examples are actually the only ones we actually care about (in some sense).

From here on out, let us agree to use the definition of a signed measure as given in the introduction. Furthermore, let us agree that it is much more desirable to work with absolutely convergent series (due to the Riemann rearrangement theorem) so that is a reasonable request to make to avoid pathologies.

Structure Theorem: Radon-Nikodym

Once we have defined some kind of object (in this case, signed measures), a natural question is the structure of said objects. Motivated by the examples above, we construct a measure that is intended to replicate the effect of the Dirac delta.

[definition] [deftitle]Definition %counter% (Dirac measure)[/deftitle]

Let $X$ be a nonempty set and fix some $a \in X$. Then we define the Dirac measure at $a$ to be the mapping

[/definition]

That is, if $X = \RR$ and $a = 0$, then

With this definition in mind, the following definition is natural comparing Lebesgue measure $m$ to the Dirac measure at $0$.

[definition] [deftitle]Definition %counter% (mutually singular)[/deftitle]

We say that the signed measure $\mu$ is singular with respect to $\nu$ if there are $E, F \in \mathcal{M}$ such that

• $E \cap F = \emptyset$;
• $E \cup F = X$;
• $E$ is $\nu$-null and $F$ is $\mu$-null.

We denote this relationship as $\mu \perp \nu$ [/definition]

[example] [extitle]Example %counter%[/extitle]

Let $\mu$ be the Dirac measure at $0$ on $\RR$ and $X = \RR$. Then we define the signed measure

where $m$ is Lebesgue measure and $S$ is Lebesgue measurable. Setting $E = 0$ and $F = \RR\setminus 0$, it follows that $\mu$ is singular with respect to $\nu$. [/example]

It turns out that, in general, if we start off with $\sigma$-finite signed measure on $(\RR, \mathcal{L})$ (where $\mathcal{L}$ is the collection of Lebesgue-measurable sets), then there exists a unique decomposition

If we drop the Delta-like part from the expression for a second and focus on the so-called “calculus-like measure”, I really just mean that

for some suitable choice of $f$ that is reasonably computable via calculus. In that sense, note that this means that $\nu(E) = 0$ whenever $E$ is Lebesgue-null. This immediately motivates the following definition.

[definition] [deftitle]Definition %counter% (absolutely continuous (measures))[/deftitle]

Let $\nu$ be a signed measure and $\mu$ a measure on the measurable space $(X, \mathcal{M})$. We say that $\nu$ is absolutely continuous with respect to $\mu$ provided that

for any $E \in \mathcal{M}$. We denote this relationship by $\nu \ll \mu$. [/definition]

[example] [extitle]Remark %counter%[/extitle]

No. I also have no idea why we use “$\ll$” as the notation for absolutely continuous. [/example]

Before we state a general version of the Radon-Nikodym theorem that is given by Folland, we will instead give a special case for $\RR$ (since it can more easily be related to calculus that way) as follows.

[theorem] [thmtitle]Theorem %counter% (Radon-Nikodym: special case, [Folland, Theorem 3.8])[/thmtitle]

Let $\nu$ be a $\sigma$-finite signed measure on $(\RR, \mathcal{L})$. Then there exists unique $\sigma$-finite signed measures $\lambda$ and $\rho$ on $(\RR, \mathcal{L})$ such that

• $\lambda \perp m$;
• $\rho \ll m$
• $\nu = \lambda + \rho$.

Furthermore, there is a function $f:\RR \to \RR$, with positive or negative part integrable, such that

for every $E \in \mathcal{L}$ and $f$ is Lebesgue a.e.-unique. [/theorem]

To make this statement more general, we replace Lebesgue measure $m$ with any $\sigma$-finite measure. This gives us the much more abstract (and impressive!) following statement.

[theorem] [thmtitle]Theorem %counter% (Radon-Nikodym, [Folland, Theorem 3.8])[/thmtitle]

Let $(X, \mathcal{M})$ be a measurable space. If $\nu$ is a $\sigma$-finite signed measure and $\mu$ a $\sigma$-finite positive measure on $(X, \mathcal{M})$, then there exist unique $\sigma$-finite signed measures $\lambda$ and $\rho$ on $(X, \mathcal{M})$ such that

• $\lambda \perp \mu$;
• $\rho \ll \mu$
• $\nu = \lambda + \rho$.

Furthermore, there is a function $f:X \to \RR$, with positive or negative part integrable such that

for every $E \in \mathcal{M}$ and $f$ is $\mu$-a.e.-unique. [/theorem]

[example] [extitle]Example %counter%[/extitle]

To keep things really concrete, let:

• $(X, \mathcal{M}) = (\RR, \mathcal{L})$ and $\mu = m$ be Lebesgue measure;
• $\lambda$ be Dirac measure at $0$ weighted by some function $g$; and
• $\rho(E) \coloneqq \int_E f\dd{m}$ for all $E \in \mathcal{L}$, where $f(x) = \sin(x)$.

Then this basically more or less amounts to the statement:

for some $a < 0 < b$. Back in physicist land, we might say this is just the integral

[/example]

There is also a natural extension of this theorem to complex measures (of interest due to Fourier theory).

References

• [Folland] Real Analysis: Modern Techniques and Their Applications by Folland.