Notation involving the Lebesgue integral. I have a measurable function $f : \mathbb{R}^d \to \mathbb{R}$. Let $E$ be a measurable subset of $\mathbb{R}^d$. Then then $$\int_{E} f(x) \, dx = \int f(x) \chi_E (x) \, dx.$$
If we are taking an integral over $\mathbb{R}^d$, shouldn't we have multiple integrals? Is this just a short hand notation that is never explicitly mentioned?
 A: The Lebesgue integral of a function $f$ on a measure space $X$ with measure $m$ is denoted
$$
\int_X f\, dm\text{ or }\int_X f(x)\,dm(x) \text{ or (especially in probability theory) }\int_X f(x) m(dx).
$$
If it is understood which measure is involved, it may be denoted
$$
\int_X f(x) \, dx.
$$
If the space happens to be $\mathbb R^d$, that is no exception.  Perhaps what is "never explicitly mentioned" is that $\mathbb R^d$ not an exception to this way of writing integrals.  What is explicitly mentioned would depend on what source you're reading.
Notice that an integral with respect to $d$-dimensional Lebesgue measure is not an integral of an integral with respect to lower-dimensional measures.  It is defined by means of measures on $d$-dimensional space, not by measures on lower-dimensional spaces.
However, it is nonetheless called a multiple (or double, or triple, etc., as the case may be) integral, as distinguished from an iterated integral.
One may write
$$
\int_{\mathbb R^2} f(x,y)\,d(x,y) \tag 1
$$
for the integral with respect to $2$-dimensional Lebesgue measure, a "double integral", or
$$
\int_{\mathbb R} \int_{\mathbb R} f(x,y)\,dx\,dy
$$
for the "iterated" (as opposed to "double") integral, which is
$$
\int_{\mathbb R} \left( \int_{\mathbb R} f(x,y)\,dx\right)\,dy,
$$
an integral of an integral, each with respect to $1$-dimensional Lebesgue measure.
Instead of $(1)$, one might denote the pair $(x,y)$ by a single letter $w=(x,y)$, and write the integral as
$$
\int_{\mathbb R^2} f(w)\,dw.
$$
A: It's like $\int_{E_1\times\cdots \times E_n} f(x_1,\ldots,x_n) \, d(x_1)\cdots d(x_n)$ where $E_1\times\cdots \times E_n=E$. Here you can use Tonelli's and Fubini's theorems.
