# What (if any) was known about null sets before Lebesgue?

The notion af a null set, i. e., a set of Lebesgue measure zero, does not require a full blown construction of Lebesgue measure:

A set is $E\subset \mathbb{R}$ is called a null-set if it can be covered by countably many intervals whose total length is as small as we please.

Many theorems and applications of measure theory (as in this question and some answers thereto), also only require the notion of null sets, possibly together with the statement that $\mathbb{R}^n$ is not a null set. And apparently, the concept appeared in the literature well before Lebesgue: according to these lecture notes, Riemann already knew what is known as "Lebesgue's characterisation of Riemann integrable functions", and thus, implicitly, the notion of null sets.

How familiar were the mathematicians with the concept of null-sets before Lebesgue? E. g., was the result of Riemann widely known? Who was the first to prove (or use in the proof) that $\mathbb{R}^n$ is not a null set? What (if any) were some other pre-Lebesgue results that used the concept of null-sets?

• Perhaps a better fit for History of Science and Math ... see the paper of Smith mentioned here: hsm.stackexchange.com/a/7375/229 – Gerald Edgar Aug 21 '18 at 17:34
• @GeraldEdgar, very interesting. I couldn't find the master thesis that you mentioned, is it available? – Kostya_I Aug 22 '18 at 14:13