Let
H
\mathcal {H}
be a Hilbert space of analytic functions on the open unit disc
D
\mathbb {D}
such that the operator
M
ζ
M_{\zeta }
of multiplication with the identity function
ζ
\zeta
defines a contraction operator. In terms of the reproducing kernel for
H
\mathcal {H}
we will characterize the largest set
Δ
(
H
)
⊆
∂
D
\Delta (\mathcal {H}) \subseteq \partial \mathbb {D}
such that for each
f
,
g
∈
H
f, g \in \mathcal {H}
,
g
≠
0
g \ne 0
the meromorphic function
f
/
g
f/g
has nontangential limits a.e. on
Δ
(
H
)
\Delta (\mathcal {H})
. We will see that the question of whether or not
Δ
(
H
)
\Delta (\mathcal {H})
has linear Lebesgue measure 0 is related to questions concerning the invariant subspace structure of
M
ζ
M_{\zeta }
. We further associate with
H
\mathcal {H}
a second set
Σ
(
H
)
⊆
∂
D
\Sigma (\mathcal {H}) \subseteq \partial \mathbb {D}
, which is defined in terms of the norm on
H
\mathcal {H}
. For example,
Σ
(
H
)
\Sigma (\mathcal {H})
has the property that
|
|
ζ
n
f
|
|
→
0
||\zeta ^{n}f|| \to 0
for all
f
∈
H
f \in \mathcal {H}
if and only if
Σ
(
H
)
\Sigma (\mathcal {H})
has linear Lebesgue measure 0. It turns out that
Δ
(
H
)
⊆
Σ
(
H
)
\Delta (\mathcal {H}) \subseteq \Sigma (\mathcal {H})
a.e., by which we mean that
Δ
(
H
)
∖
Σ
(
H
)
\Delta (\mathcal {H}) \setminus \Sigma (\mathcal {H})
has linear Lebesgue measure 0. We will study conditions that imply that
Δ
(
H
)
=
Σ
(
H
)
\Delta (\mathcal {H}) = \Sigma (\mathcal {H})
a.e.. As one corollary to our results we will show that if dim
H
/
ζ
H
=
1
\mathcal {H}/\zeta \mathcal {H} =1
and if there is a
c
>
0
c>0
such that for all
f
∈
H
f \in \mathcal {H}
and all
λ
∈
D
\lambda \in \mathbb {D}
we have
|
|
ζ
−
λ
1
−
λ
¯
ζ
f
|
|
≥
c
|
|
f
|
|
||\frac {\zeta -\lambda }{1-\overline {\lambda }\zeta }f||\ge c||f||
, then
Δ
(
H
)
=
Σ
(
H
)
\Delta (\mathcal {H}) =\Sigma (\mathcal {H})
a.e. and the following four conditions are equivalent: (1)
|
|
ζ
n
f
|
|
↛
0
||\zeta ^{n} f||\nrightarrow 0
for some
f
∈
H
f \in \mathcal {H}
, (2)
|
|
ζ
n
f
|
|
↛
0
||\zeta ^{n} f||\nrightarrow 0
for all
f
∈
H
f \in \mathcal {H}
,
f
≠
0
f \ne 0
, (3)
Δ
(
H
)
\Delta (\mathcal {H})
has nonzero Lebesgue measure, (4) every nonzero invariant subspace
M
\mathcal {M}
of
M
ζ
M_{\zeta }
has index 1, i.e., satisfies dim
M
/
ζ
M
=
1
\mathcal {M}/\zeta \mathcal {M} =1
.