In Cohn's measure theory, second edition, p166, there is written:
Let $\mu$ be a finite Borel measure on $\mathbb{R}^d$. Then the upper derivate $(\overline{D}\mu)(x)$ of $\mu$ at $x$ is defined by
$$(\overline{D}\mu)(x) = \lim_{\epsilon \to 0}\sup \left\{\frac{\mu(C)}{\lambda(C)}: C \in \mathcal{C}, x \in C, e(C) < \epsilon\right\}$$
where $\mathcal{C}$ is the set of all cubes in $\mathbb{R}^d$ and $e(C)$ denotes the length of a side of a cube.
Can someone explain what the notation $(\overline{D}\mu)(x) = \lim_{\epsilon \to 0}\sup \left\{\frac{\mu(C)}{\lambda(C)}: C \in \mathcal{C}, x \in C, e(C) < \epsilon\right\}$ means?
I am only familiar with $\limsup$ of sequences, for example $\limsup_{n \to \infty} x_n = \inf_{n} \sup_{k \geq n} x_k$
Best Answer
This Community Wiki answer (I'll add more to it later) can be used to make a list of places where suitable definitions of $\limsup$ can be found. (It would also be a good idea to include the relevant parts of at least some of the texts.)
D. J. H. Garling, A Course in Mathematical Analysis, Vol. I (Cambridge University Press 2013), p.150f.
J. M. Hyslop, Real Variable (Oliver & Boyd 1960), p.107f.
Brian S. Thomson, Judith B. Bruckner & Andrew M. Bruckner, Elementary Real Analysis: DRIPPED Version (2008):
The book is freely downloadable. It may have been updated since I downloaded it. The definition needs to be adapted to the case of a one-sided limit as $\epsilon\to0$: this is exercise 5.3.6 in the book.