In many physics books (especially QFT), the integral measure for a quantum field is written down as
\begin{equation} \phi(x)=\int\frac{d^{3}p}{(2\pi)^{3}}.... \end{equation}
It is a summation over the continuous momentum variable in 3 (Euclidian) space. However, in these same books, 3-vectors are distinguished by bold Latin letters $\bf{p}$. shouldn't the integral measure instead be written as \begin{equation} \int\frac{d^{3}\bf{p}}{(2\pi)^{3}}? \end{equation}
But then, it could imply that the quantum field $\phi(x)$ is a vector quantity (which isn't). For example in Stoke's divergence theorem, the integral $d\bf{a}$ is a 3-vector quantity (area element with orientation). what is the best way out of this confusion, i.e. what is the correct mathematical way to express the integral measure in quantum fields?