42 1. Higher order Fourier analysis
measure on [N]. (Hint: Use the Carath´ eodory extension theorem,
see e.g. [Ta2011, §1.7].)
(iii) If f : [N] C is a limit function bounded in magnitude by some
standard real M, show that st(f) is a Loeb measurable function in
L∞(μ),
with norm at most M.
(iv) Show that there exists a unique trilinear form Λ:
L∞(μ)×L∞(μ)×
L∞(μ)
C, jointly continuous in the
L3(μ)
topology for all three
inputs, such that
Λ(st(f), st(g), st(h))
= st(
1
N 2
n∈[N] r∈[−N,N]
f(n)g(n + r)h(n + 2r))
for all bounded limit functions f, g, h.
(v) Show that Roth’s theorem is equivalent to the assertion that
Λ(f, f, f) 0 whenever f
L∞(μ)
is a bounded non-negative
function with
[N]
f 0.
Loeb measure was introduced in [Lo1975], establishing a link between stan-
dard and non-standard measure theory.
Next, we develop the ultralimit analogue of Fourier measurability, which
we will rename Kronecker measurability due to the close analogy with the
Kronecker factor in ergodic theory.
Exercise 1.2.14 (Construction of the Kronecker factor). Let N be an un-
bounded natural number. We define a Fourier character to be a function in
L∞([N])
of the form n st(e(αn)) for some limit real number α. We define
a trigonometric polynomial to be any finite linear combination (over the
standard complex numbers) of Fourier characters. Let
Z1
be the σ-algebra
of Loeb measurable sets generated by the Fourier characters; we refer to
Z1
as the Kronecker factor, and functions or sets measurable in this factor as
Kronecker measurable functions and sets. Thus, for instance, all trigono-
metric polynomials are Kronecker measurable. We let
E(f|Z1)
denote the
orthogonal projection from f to
L2(Z1),
i.e., the conditional expectation to
the Kronecker factor.
(i) Show that if f
L∞(Z1)
is bounded in magnitude by M and ε 0
is a standard real, then there exists a trigonometric polynomial
P
L∞(Z1)
which is also bounded in magnitude by M and is
within ε of f in
L1
norm.
(ii) Show that if f
L∞(Z1)
and ε 0, then there exists a limit
subset R of [−εN, εN] of cardinality N such that f(·) f(· +
r)
L1([N])
ε for all r R (extending f by zero).
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