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 dμ 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).