Is the continuous image of a Borel subset Lebesgue measurable?

Question

Let $f:\mathbb R^n \to \mathbb R^n$ be a continuous map and $B \subset \mathbb R^n$ a Borel subset. It is well known that $f(B)$ may not be a Borel subset.

My question is, can we prove that $f(B)$ must be measurable for Lebesgue measure?

Answer 1

My question is, can we prove that f(B) must be measurable for Lebesgue measure?

The answer to your question is, of course, ...

Mathematics students don't want or need answers, certainly not a simple affirmative or negative.

Mathematics is a human endeavor and the stories are what interest us. For this question the story is really very interesting. Imagine you are a young Russian student working in Moscow in 1914 with the great Lusin. He assigns you a paper to read by the incomparable and, by then, hugely famous French mathematician Lebesgue" Sur les functions representables analytiquement (1905).

Lebesgue has "proved" that the projection of a Borel set must be a Borel set and obtained some results using that. But it is a mistake. More correctly, it is a blunder. Now you, a mere second year undergraduate student, are going to rush off to the great master Lusin and tell him "Lebesgue has made a mistake. I have a counterexample."

Within a year of working with his very supportive professor Lusin, Suslin had characterized the class of sets that are continuous images of Borel sets. He analysed them using an operation due to Alexandrov, from another paper that Lusin had assigned to him. Suslin introduced this class of A-sets (named after Alexandrov) as an extension of B-sets (as Borel sets were often then called).

His work completely characterizes the family of sets in your problem: those sets that are continuous images of Borel sets. His characterization reveals that this forms a strictly larger class than the Borel sets and all such sets are Lebesgue measurable. [Answer to your question?]

Suslin died of typhus on 21 December 1919 at the age of 23.

According to Alexandrov "Until the end of Lusin's life a portrait of Suslin stood on his desk, the only portrait of Suslin that I have seen."