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Part of 2011 Miklós Schweitzer

Problems(1)

borel sets

Source: miklos schweitzer 2011 q1

8/29/2021
Let F1,F2,...F_1, F_2, ... be Borel-measurable sets on the plane whose union is the whole plane. Prove that there is a natural number n and circle S for which the set S∩FnS \cap F_n is dense in S. Also show that the statement is not necessarily true if we omit the condition for the measurability of sets FjF_j.
Measure theorytopology