The following result, which I know under the name Fekete’s lemma
is quite often useful. It was, for example, used in this answer: Existence of a limit associated to an almost subadditive sequence.
If $(a_n)_{n=0}^\infty$ is a subadditive sequence of real numbers, i.e.,
$$(\forall m,n) a_{m+n} \le a_m + a_n,$$
then $$\lim\limits_{n\to\infty} \frac{a_n}n = \inf_n \frac{a_n}n.$$
Some references are given in Wikipedia article, the original Fekete’s paper is available here.
Basically the exponential version (for submultiplicative sequences) can be shown in a similar way as Satz II in this paper.
I was wondering, whether some analogous claim is true for functions. I.e. something like: Whenever $f:{(0,\infty)}\to{\mathbb R}$ fulfills
$$(\forall x,y)f(x+y) \le f(x)+f(y)$$
(i.e., it is subadditive), then
$$\lim\limits_{x\to\infty} \frac{f(x)}x = \inf_x \frac{f(x)}x.$$
(In particular, the above limit exists — if we include the possibility $-\infty$.)
Clearly, this is not true without any additional assumptions on $f$. (E.g. if $f$ is any non-linear solution of Cauchy’s equation, then $\liminf \frac{f(x)}x < \limsup \frac{f(x)}x$ and $f$ is both subadditive and superadditive. Probably even much simpler examples can be given.)
On the other hand, if $f$ is well-behaved, the above claim is true. If I assume that $f$ is bounded on intervals of the form $(0,x]$, then I can basically repeat the proof which is given for sequences here.
So my question is:
Under what assumptions on $f$ the above claim is true.
Can you give some references for this claim?
EDIT: I found a result which shows that measurability of $f$ is sufficient and added this result as an answer. I think this is sufficient for most applications and my guess is that there is not much space to improve this result. However, I will wait a little bit before accepting my own answer – just in case someone would like to add some interesting information or further useful references. I have accepted my own answer, but if you have some interesting information which you can add, I’ll be very glad to learn about it.
If found the following $N$-dimensional result in the book An introduction to the theory of functional equations and inequalities By Marek Kuczma p.463:
Theorem 16.2.9. Let $f:\mathbb R^{N}\to\mathbb R$ be a measurable subadditive function. Then for every $x\in\mathbb R^N$ there exists the limit $$F(x)=\lim_{t\to\infty} \frac{f(tx)}t.$$ The function $F$ is finite, continuous in $\mathbb R^N$, positively homogeneous and subadditive.
I should also mention that in the proof of this theorem it is shown that
$$\lim_{t\to\infty} \frac{f(tx)}t=\inf_{t>0}\frac{f(tx)}t.$$
This result is proven in Kuczma’s book and he gives the following texts as further references:
E. Hille and R. S. Phillips, Functional analysis and semi-groups, American Mathematical Society Colloquium Publications, vol. 31, American Mathematical Society, Providence, R. I., 1957, rev. ed. Special case for $N=1$ is given in this book as Theorem 7.6.1. Here the assumptions are that $f$ is a real subadditive function defined on some interval $(a,\infty)$, $a\ge 0$.
R.A. Rosenbaum, Sub-additive functions, Duke Math. J. 17 (1950), 227–247.
I also stumbled upon the paper J.M. Hammersley: Generalization of the Fundamental Theorem on Subadditive Functions, where the author refers to this result as fundamental theorem on subadditive functions.
This shows that measurability of $f$ is sufficient for Fekete’s lemma to hold.