Let
Γ
∈
[
0
,
1
]
\Gamma \, \in \,[0,\,1]
be Lebesgue measurable; then
Γ
\Gamma
has Lebesgue density 0 at the origin if and only if
\[
∫
Γ
t
−
1
Ψ
(
t
−
1
meas
{
Γ
∩
(
0
,
t
)
}
)
d
t
>
∞
\int _\Gamma {{t^{ - 1}}\Psi ({t^{ - 1}}\,{\text {meas}}} \{ \Gamma \, \cap \,(0,\,t)\} )\,dt\, > \,\infty
\]
for some continuous, strictly increasing function
Ψ
(
t
)
(
0
⩽
t
⩽
1
)
\Psi (t)\,(0\, \leqslant \,t\, \leqslant \,1)
with
Ψ
(
0
)
=
0
\Psi (0)\, = \,0
. This result is applied to the local growth of certain Gaussian (and other) proceses
{
X
t
,
t
⩾
0
}
\{ {X_t},\,t\, \geqslant \,0\}
as follows: we find continuous, increasing functions
ϕ
(
t
)
\phi (t)
and
η
(
t
)
(
t
⩾
0
)
\eta (t)\,(t\, \geqslant \,0)
such that, with probability one, the set
{
t
:
η
(
t
)
⩽
|
X
t
−
X
0
|
⩽
ϕ
(
t
)
}
\{ t:\eta (t)\, \leqslant \,\left | {{X_t}\, - \,{X_0}} \right |\, \leqslant \,\phi (t)\}
has density 1 at the origin.