Ham sandwich theorem. Recently, I have been reading the Borsuk-Ulam theorem from Hatcher's algebraic topology book. In this book, there is an exercise as follows.
Let $A_1, A_2, A_3$ be compact sets in$\mathbb{R}^3$. Use the Borsuk–Ulam theorem to show that there is one plane $P \subset \mathbb{R}^3$ that simultaneously divides each $A_i$ into two pieces of equal measure.
I have some doubts regarding the above statement; each $A_i$ divided equal measure this part I did not understand properly. $\mathbb{R}^3$ has its won Lebesgue measure say $\mu$ if I take one of $A_i$ suppose $A_1$ say a unit circle $S^1$ which is compact and $\mu(S^1)=0$ then no need to divided by two equal measure. Is this statement say when I am taking one of them as the lesser dimension, we must take the Lebesgue measure concerning that dimension?
Can anyone suggest books or good references where I can find the statement of the ham sandwich theorem and its proofs?

I can only recommend the 3b1b video on the Stolen Necklace Problem, which explains the visual intuition behind a very similar problem (though you'd need to formalize it a bit and adapt it to your situation).
(You can skip everything from 3:45 to 10:45, which is just explaining the Borsuk-Ulam theorem. The part you really care about is around the 16:00 timecode, but it's good to see the statement of the problem first)
