When we integrate with respect to some measure $\mu$ we write $\int f\, d\mu$ or $\int f(x)\, d\mu (x)$. For example for Lebesgue measure we write $\int fd\lambda$, to emphasize we mean $n$ dimensional Lebesgue measure we use $\lambda_n$. When we do Riemann Integral we use $dx$, for example $\int_0^1x^2dx$. What measure is this? I mean "$dx$" Does it have a name?
Suppose we want to calculate Lebesgue measure of set $A=\{(x,y) \in R^2 : x+y \leq 1\}$. Using Fubini theorem we write:
$$\int_Ad\lambda_2=\int_0^1\int_0^{1-x}\, dydx$$
What are names of these measures $dy$ and $dx$? Are they Lebesgue measure? Are $dx$ and $d\lambda_1$ the same symbols?
