Measurable Rectangles 1 Arnold

We give an example of a measurable set E ⊆ R such that the set E ′ = {(x, y) : x + y ∈ E} is not in the σ-algebra generated by the rectangles with measurable sides. We also prove a stronger result that there exists an analytic (Σ1) set E such that E ′ is not in the σ-algebra generated by rectangles whose horizontal side is measurable and vertical side is arbitrary. The same results are true when measurable is replaced with property of Baire. The σ-algebra generated by a family F of subsets of a set X is the smallest family containing F and closed under taking complements and countable unions. In Rao [12] it is shown that assuming the Continuum Hypothesis every subset of the plane R is in the σ-algebra generated by the abstract rectangles, i.e. sets of the form A × B where A and B are arbitrary sets of reals. In Kunen [5] it is shown that it is relatively consistent with ZFC that not every subset of the plane is in the σ-algebra generated by the abstract rectangles. He shows that this is true in the Cohen real model. It also follows from a result of Rothberger [14] that if for example 2א0 = א2 and 2א1 = אω2 , then not not every subset of the plane is in the σ-algebra generated by the abstract rectangles. For a proof of these results see Miller [11] (remark 4 and 5 page 180). A set is analytic or Σ1 iff it is the projection of a Borel set. Answering a question of Ulam, Mansfield [7][8] showed that not every analytic subset of the plane is in the σ-algebra generated by the analytic rectangles. Note that a rectangle A×B ⊆ R× R is analytic iff both A and B are analytic. He did this by showing that, in fact, any universal analytic set is not in the σ-algebra generated by the rectangles with measurable sides. This does the trick because analytic sets are measurable (see Kuratowski [6]). This theorem was also proved by Rao [13]. Their argument shows a little more, in Real Analysis Exchange, 19(1994), 194-202. Research partially supported by NSF grant DMS-9024788.