Probability Measure - Riemann vs Lebesgue There is something causing me trouble about Riemann and Lebesgue integrals. There are a lot of articles that define some sort of probability measure as follows:
$d\mu(x) = f(x,y)dxdy$
This anooys me because I don't know if the use of "$dxdy$" is just a matter of notation abuse or if it really means an infinitesimal element of Riemann-integral, which I consider not to be any kind of measure in the sense of measure theory. So, I came to the conclusion that one may define this so called "probability measure" if the function f(x,y) is Riemann-Integrable onde you may exchange integrals in the latter case but the formal definition would be using Lebesgue measure indeed. 
Am I right?
 A: Let $\lambda$ denote the Lebesgue measure on $\mathbb{R}^n$ for $n \in \mathbb{N}$. Take any $\sigma$-finite measure $\mu$ on the same measurable space such that for any measurable set $A$ we have that $\lambda(A)=0$ implies that $\mu(A)=0$. We say that $\mu$ is absolutely continuous with respect to $\lambda$. Radon-Nikodym theorem implies that there exist a measurable function $f$ such that
$$(\star) \ \ \quad  \mu(A) = \int_A f(x) \ \lambda(dx).$$
Since $\mu(A) = \int_A \mu(dx)$ we often abbreviate $(\star)$ by writing
$$ \mu(dx) = f(x) \ \lambda(dx) \ \mbox{ or just } \ d\mu(x) = f(x) \ d\lambda(x).$$
We call $f$ the Radon–Nikodym derivative or the density and sometimes denote $\frac{d\mu}{d\lambda}$. Often we use the following notation for the Lebsgue measure:


*

*if $n=1$ then we simply denote $\int \lambda(dx) = \int dx$ and even further we denote $\lambda(dx)$ as $dx$, 

*if $n=2$ then we simply denote $\int \lambda(d(x,y)) = \int d(x,y) = \int \ dxdy$ and even further we denote $\lambda(d(x,y))$ as $d(x,y)$ or $dxdy$ etc.


Of course Lebesgue measure is not a probability measure on $\mathbb{R}^n$, but you can adapt the above for probability measures as well, e.g. Lebesgue measure on $[0,1]^n$.
