TITLE: Strange definitions about basic probability - need clarification
QUESTION [7 upvotes]: In the book Condition. The Geometry of Numerical Algorithms - Peter Burgisser, Felipe Cucker, there is an explanation that doesn't make sense to me. It's just a problem about definitions, but I really need a light on this. This section of the book is supposed to be a review of probability, so it's not necessary to know the book to answer my question, just some good knowledge of probability theory. 
Below is the page from the book which I'm having trouble with.

The first problem comes from the integral $\int_M f = 1$, which makes part of the definition of $f$. This integral has to be defined in respect to some measure over $M$ (more precisely, there is a measure $\mu$ over $M$ so that $\int_M f = \int_M f\ d\mu$) . Are they assuming this definition makes sense for any measure over $M$? Or maybe already there is some probability measure over $M$?  
My second problem is: after the definition of density, they say we can use $f$ to define a probability measure $\mu$ on $M$ with another integral, but this another integral doesn't seen to depend on $f$. Where $f$ is hidden in this definition? And again, we are integrating in respect to what measure?
Finally, this is just an observation, but I need some clarification on this too. Usually we work with sample space $\Omega$ instead data space $M$. I'm assuming this is just a mater of conventions, so this is ok. But as far as I know, usually we work with some random variable $X:\Omega\to S$ ($S$ is a Polish space) and the density is a function $f:S\to[0,\infty]$. The way they are defining the density (before mention random variables) makes it be $f:\Omega\to[0,\infty]$. Honestly, I never saw a density defined directly on the sample space. Is this a normal routine?
Thank you and sorry for the long post.

REPLY [2 votes]: I would say a density $f$ defines a probability measure $P$ via the integral $\displaystyle P(A) = \int_A f$.
Omitting even to mention $f$ in this statement makes no sense to me.
Where the book says $\displaystyle\int_M \mathbf 1_A$ it should have said $\displaystyle \int_M \mathbf 1_A \cdot f$ or $\displaystyle P(A) = \int_A f$.
To avoid confusion resulting from the use of the same symbol for several different things in the question and the answers, I will use the letter $m$ for a measure on $M$.  The integral above is in fact an integral with respect to a measure $m$ on $M$.  So
$$
P(A) = \int_A f\,dm.
$$
One says that $f$ is the density of the probability measure $P$ with respect to $m$.  The measure $m$ need not be a probability measure.  One can also express this by saying $f = \dfrac{dP}{dm}$, and then one says $f$ is the Radon–Nikodym derivative of $P$ with respect to $m$.