What is the difference between $\int f(x)dx$ and $\int f(x) dm(x)$?
In an exercise I was asked to compute $\int\limits_{[0,1]\times[0,1]}xe^{xy}dm(x,y)$ and I did it using the definition $\int f(x,y)dm(x,y)=\int\limits_0^\infty m(\{(x,y): f(x,y)>t\})dt$ but then the TA told me that $dm(x,y)$ is the same as $d\vec{(x,y)}$ and by using Tonelli (since $f(x,y)>0$ in this case) the integral is simply $\int\limits_0^1\int\limits_0^1 f(x,y)dydx$
So my question is: why do we use two different notations for the same thing? And what exactly is $dm(x,y)$?
The map $(x,y)\mapsto xe^{xy}$ on $[0,1]^2$ is nonnegative, so by Tonelli's theorem, $$ \iint_{[0,1]^2} xe^{xy}\ \mathsf dm(x,y)<\infty $$ and is equal to the integrated integrals $$ \int_0^1\int_0^1 xe^{xy}\ \mathsf dx\ \mathsf dy $$ and $$ \int_0^1\int_0^1 xe^{xy}\ \mathsf dy\ \mathsf dx. $$ In general, $\mathsf d m(x,y)$ means integrating with respect to two-dimensional Lebesgue measure (in this case restricted to the cube $[0,1]^2$). Meanwhile, the iterated integrals are integration with respect to one-dimensional Lebesgue measure, performed twice in a row. These are in general not the same thing, but coincide under the assumptions of Tonelli's (or Fubini's) theorem.