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".

  • 1
    $\begingroup$ "Base measure" doesn't have any special meaning. In this context it just means "the measure that all the functions in $\mathcal{F}$ are densities with respect to". $\endgroup$ – Nate Eldredge Apr 23 at 16:52
  • $\begingroup$ And what is the mathematical characteristics of "the measure that all the functions in $\cal F$ are densities with respect to"? Like "the integrals over the support of the base measure is equal to 1"? $\endgroup$ – user25004 Apr 23 at 17:52
  • $\begingroup$ It has no special mathematical characteristics, it could be any measure. It's just a word attached to the particular measure that you happen to be working with here. It's similar to a sentence like "the set of all paths $\gamma$ for which $\gamma(0)$ is the starting point $x$". "Starting" isn't some property of points, it's just a name for a particular point where all our paths start. Likewise, "base" isn't some property of measures, it's just a name for a particular measure that we're using as context for the functions in $\mathcal{F}$. $\endgroup$ – Nate Eldredge Apr 23 at 17:56

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