What does this notation mean: $\limsup_{\epsilon \to 0} \dots$

Question

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 $\limsup_{n \to \infty} x_n = \inf_{n} \sup_{k \geq n} x_k$

Answer 1

Are you sure they really mean the $\limsup$? To me, it looks more like one is supposed to take the supremum of the set and then pass to the limit, as it says $\lim_{\epsilon\to0}\sup$ and not $\limsup_{\epsilon\to0}$.

In very small steps: Look at the set $$ A_\epsilon:= \left\{\frac{\mu(C)}{\lambda(C)}: C \in \mathcal{C}, x \in C, e(C) < \epsilon\right\}, $$ take its supremum $$ S_\epsilon:= \sup A_\epsilon \in\mathbb [0,\infty],$$ and then pass to the limit $$ \lim_{\epsilon \to 0}S_\epsilon, $$ which has to exist in $[0,\infty]$, as by the definition of $A_\epsilon$, the quantity $S_\epsilon$ is decreasing for $\epsilon\to0$.

Answer 2

The limsup of a family of sets $D_\epsilon$ is defined as $$\limsup_{\epsilon \to 0+} D_\epsilon = \bigcap_{\epsilon > 0}\bigcup_{\epsilon' < \epsilon} D_\epsilon.$$ This can be thought of as the set of the elements $x$ such that whatever $\epsilon_0 > 0$, there exists a $\epsilon$ such that $x \in D_\epsilon$ or the elements that are in infinitely many $D_\epsilon, \epsilon < \epsilon_0$ for any $\epsilon_0 >0$ .

Answer 3

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.)

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D. J. H. Garling, A Course in Mathematical Analysis, Vol. I (Cambridge University Press 2013), p.150f.

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J. M. Hyslop, Real Variable (Oliver & Boyd 1960), p.107f.

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Brian S. Thomson, Judith B. Bruckner & Andrew M. Bruckner, Elementary Real Analysis: DRIPPED Version (2008):

5.3 Limits Superior and Inferior

If limits fail to exist we need not abandon all hope of discussing the limiting behavior. We saw this situation in our study of sequence limits in Section 2.13. Even if $\{s_n\}$ diverges so that $\lim_{n\to\infty}s_n$ fails to exist, it is possible that the two extreme limits $$ \liminf_{n\to\infty}s_n \text{ and } \limsup_{n\to\infty}s_n $$ provide some meaningful information. These two concepts always exist (possibly as $\infty$ or $-\infty$). A similar situation occurs for functions. The theory is nearly identical in many respects.

Definition 5.26: (Lim Sup and Lim Inf) Let $f \colon E \to \mathbb{R}$ be a function with domain $E$ and suppose that $x_0$ is a point of accumulation of $E$. Then we write $$ \limsup_{x\to x_0}f(x) = \inf_{\delta>0}\sup \{f(x) : x \in (x_0 - \delta, x_0 + \delta), \ x \ne x_0\} $$ and $$ \liminf_{x\to x_0}f(x) = \sup_{\delta>0}\inf \{f(x) : x \in (x_0 - \delta, x_0 + \delta), \ x \ne x_0\} $$ As this section is for more advanced readers we have left the development of this concept to the exercises. [...]

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.