1000 CHAPTER 28. HAUSDORFF MEASURE

and these are Borel measurable functions of Pix. Also, if {Ai} is a disjoint sequence of setsin G then

m((∪iAi∩Rk)Pix

)= ∑

im((Ai∩Rk)Pix

)and each function of Pix is Borel measurable. Thus by the lemma on π systems, Lemma12.12.3, G = B (Rn) and this proves the lemma.

Now let A⊆ Rn be Borel. Let Pi be the projection onto

span(e1, · · · ,ei−1,ei+1, · · · ,en)

and as just described,APix = {y ∈ R : Pix+ yei ∈ A}

Thus for x = (x1, · · · ,xn),

APix = {y ∈ R : (x1, · · · ,xi−1,y,xi+1, · · · ,xn) ∈ A}.

Since A is Borel, it follows from Lemma 28.3.1 that

Pix→ m(APix)

is a Borel measurable function on PiRn = Rn−1.

28.3.1 Steiner SymmetrizationDefine

S(A,ei)≡ {x =Pix+ yei : |y|< 2−1m(APix)}

Lemma 28.3.3 Let A be a Borel subset of Rn. Then S(A,ei) satisfies

Pix+ yei ∈ S(A,ei) if and only if Pix− yei ∈ S(A,ei),

S(A,ei) is a Borel set in Rn,

mn(S(A,ei)) = mn(A), (28.3.6)

diam(S(A,ei))≤ diam(A). (28.3.7)

Proof: The first assertion is obvious from the definition. The Borel measurability ofS(A,ei) follows from the definition and Lemmas 28.3.2 and 28.3.1. To show Formula28.3.6,

mn(S(A,ei)) =∫

PiRn

∫ 2−1m(APix)

−2−1m(APix)dxidx1 · · ·dxi−1dxi+1 · · ·dxn

=∫

PiRnm(APix)dx1 · · ·dxi−1dxi+1 · · ·dxn

= m(A).