A nonnegative function on the real line satisfies the condition
A
∞
{{\mathbf {A}}_\infty }
if, given
ε
>
0
\varepsilon > 0
, there exists a
δ
>
0
\delta > 0
such that if
I
I
is an interval,
E
⊂
I
E \subset I
, and
|
E
|
>
δ
|
I
|
|E| > \delta |I|
, then
∫
E
W
≤
ε
∫
I
W
\int _E {W \leq \varepsilon \int _I W }
. A nonnegative function on the real line satisfies the condition
A
{\mathbf {A}}
if for every interval
I
,
∫
2
I
W
≤
C
∫
I
W
I,\int _{2I} {W \leq C} \int _I W
, where
2
I
2I
is the interval with the same center as
I
I
and twice as long, and
C
C
is independent of
I
I
. An example is given of a function that satisfies
A
{\mathbf {A}}
but not
A
∞
{{\mathbf {A}}_\infty }
.