Remark on measurable functions with arbitrarily small periods
Citation for published version (APA):Steen, van der, P. (1978). Remark on measurable functions with arbitrarily small periods. (Eindhoven University of Technology : Dept of Mathematics : memorandum; Vol. 7813). Technische Hogeschool Eindhoven.
Document status and date: Published: 01/01/1978 Document Version:
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I
EINDHOVEN UNIVERSITY OF TECHNOLOGY
Department of Mathematics
Memorandum 1978- 13 november 1978
Remark on measurable functions with arbitrarily small periods
University of Technology
Department of Mathematics PO Box 513, Eindhoven The Netherlands
by
Remark on measurable functions with arbitrarily small periods
by
P. van der Steen
A recent issue of the American Mathematical Monthly [
J
contains a note on the following result of Burstin (1915):Theorem. Let f be a Lebesgue measurable extended real-valued function defin -ed onm having arbitrarily smal l posi tive periods. Then f is almost constant, i.e. there exists c such that f(x) = c (a.e.).
There are a number of proofs of this result (for references see [lJ), most of which use integration in some form. In this note we present a proof using only the concepts of measure and measurable function, and the fact that Lebesgue measure ~ onm is translation invariant. We need a lemma (which must be in many places in the literature) •
Lemma. Let A be a measurable subset of m with ~(A) > O. Let V be a dense sub-set ofm. If Y := A + V := {a + v
I
a E A, v E V}, then~(
Y
*
)
= O.Proof. We may suppose that V is countable, and then Y is obviously measura
-*
ble. Assume ~(Y ) > O. Choose open intervals I and J such that
3
*
3o
< ~ (I) < ~ (J) < 2~ (I), ~ (I n A)>
"4
~ (I), ~ (J n Y )>
"4
~ (J) .Choose v E V such that I + v C J. Now
~ (Y n J) ~ ~(Y n (I + v~) = ~(Y n I) ~ ~(A n I) ~
"4
3 ~(I) >8
3 ~(J)Hence
~ (J) ~(Y
n
J) + ~(Y*
n
J)>
"4
3 ~(J) +8
3 ~(J)>
~(J) :contradiction.
The proof of Burstin's result is easy now. Let Vo be the set of periods of f. Since Vo contains arbitrarily small positive periods, Vo as a subgroup ofm must be dense inm. Let V be a countable dense subset of Vo such that if v E V, then -v E V. For each v E V there exists a null set E such that
f (x) == f (x)
v
*
f(x + v) for x E E • Let E:== U E , then E is a null set , and
v VEV v
f(x + v) for x E E*, v E V. For each q Em, let A := {XE E*
I
f(x)~q
}.
- 2
-Then A + V == A
q q and A q
*
+ V A q*
, : and the lemma shows that either )l (A q ) == 0or )l(A*) == O. q
If f(x) == 00 (a.e.) or f(x) == _00 (a.e.) we are done already. If not,
then there exist q and q' such that )leA ) == 0 and )leA ,) > O. If
q
*
qc :== sup{q
I
)l(Aq)a},
then c E ffi and )l(Aq) == 0 for q > c. Hence f(x) == c(a.e.) in this case.
Reference
[lJ R. Cignoli and J. Hounie, Functions with arbitrarily small periods,
