What is the base measure in measure theory?

Question

I see the term "base measure" used frequently about measures. I do not completely get what that exactly means:

Some examples are:

Let $\cal F$ be the space of all probability density functions with respect to a base measure $\nu$

What is the base measure?

Sometimes when a probabilistic function is integrated,

the dx is called a base measure. $$\int_{\cal X} .... dx$$

Can someone explain in simple words or refer me to a simple reference to read about "base measures".

Answer 1

The word "base" doesn't have any particular mathematical significance here, and could be omitted without changing the mathematical meaning. It doesn't signify that the measure has any particular mathematical properties. It's just being used in its common English sense of "something to build on" or "foundation", and serves as a hint that this measure is going to be used to build other stuff (i.e. densities) on top of.

So the word isn't really meant to formally affect the math itself, but rather informally, to help guide the reader's intuition about the math.

Answer 2

When I see the word "base measure", the first thing that comes to me is the Radon-Nikodym theorem (and the Radon-Nikodym derivative).

The Radon-Nikodym theorem states the following.

Consider a measurable space $(X,\Sigma )$ on which two $\sigma$-finite measures $\mu$ and $\nu$ are defined. If $\nu$ is absolutely continuous with respect to $\mu$ (i.e., $\mu (A) = 0$ implies $\nu (A) = 0$ for all measurable $A \subseteq X$), then there exists a $\Sigma$-measurable function $f:X \rightarrow [0,\infty )$, such that for any measurable set $ A \subseteq X$, $$ \nu (A)=\int _{A}f\,d\mu. $$

In addition, if two functions $f_1$ and $f_2$ satisfies the same property, then $$\mu ( \cup_{A \subseteq X, A\text{ measurable} } \{ A : f_1 (A) \neq f_2 (A) \} ) = 0.$$

In words, we say such $f$ is $\mu$-almost everywhere unique. Because of this uniqueness one often write $ f = \frac{ d \nu }{ d \mu } $ (provided that $\nu$ is absolutely continuous with respect to $\mu$). In such cases, people often call $\mu$ the base measure.

For me (who mostly works in Euclidean spaces and rarely does research in measure theory), I think of the Lebesgue measure and assume that a density function can be properly written out when I hear the word base measure.