Fekete's lemma for real functions

Question

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.

Answer 1

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.