Basically, I'm not a mathematician, and I'm trying to figure out how to be "precise" in terms of notation and logic for these concepts.
Say I have a function $\lambda(\mathbf{x}) \geq 0$, defined on some $\mathbf{x} \in \mathbb{R}^n$, which is "nice." I define the measure (this would be a measure right?):
$\Lambda (B) = \int _{B} \lambda(\mathbf{x}) d\mathbf{x}$
where we are considering compact (?) domains $B \subset \mathbb{R}^n$. If $\lambda$ is continuous over $B$, it seems obvious to me to state that as B becomes arbitrarily small, $\Lambda (B)$ will become $\lambda (\mathbf{x}_B)|B|$, where I'm saying $\mathbf{x}_B$ is some point in $B$.
How would I state this as an equation where I'm taking the limit? Like what is the correct notation for taking the limit of B as it becomes an infinitesimally small volume at $\mathbf{x}_B$? Also, what are the conditions on $\lambda$ for this work? Just that it has to be continuous?