Analogue of natural densitiy on positive real numbers

Question

There is the following analague of natural density on the positive real numbers: Suppose that $A\subseteq (0,\infty)$ is Lebesgue-measurable, then one can look at the limit (if it exists) $\frac{\lambda(A\cap(0,n))}{n}$ for $n\rightarrow\infty$, where $\lambda$ is the usual Lebesgue measure on $(0,\infty)$. My question is, if there exists a name for this concept, something like "real density" maybe instead of natural density?

Answer 1

Perhaps counterintuitively, the word "natural" in "natural density" doesn't refer to the natural numbers. Rather, it refers to the fact that the density is the most natural/obvious density one would define; it contrasts with other densities like the logarithmic density $$ \lim_{x\to\infty} \frac1{\log x} \int_{A\cap[1,x]} \frac{dt}t. $$ In light of this, people typically call the density defined in the OP the "natural density" on the positive real numbers.