Let
α
:
[
0
,
∞
)
→
C
\alpha :[0,\infty ) \to {\mathbf {C}}
be a function of locally bounded variation, with
α
(
0
)
=
0
\alpha (0) = 0
, whose Laplace-Stieltjes transform
g
(
z
)
=
∫
0
∞
e
−
z
t
d
α
(
t
)
g(z) = \int _0^\infty {{e^{ - zt}}d\alpha (t)}
is absolutely convergent for
Re
z
>
0
\operatorname {Re} z > 0
. Let
E
E
be the singular set of
g
g
in
i
R
i{\mathbf {R}}
, and suppose that
0
∉
E
0 \notin E
. Various estimates for
lim
sup
t
→
∞
|
α
(
t
)
−
g
(
0
)
|
\lim {\sup _{t \to \infty }}|\alpha (t) - g(0)|
are obtained. In particular,
α
(
t
)
→
g
(
0
)
\alpha (t) \to g(0)
as
t
→
∞
t \to \infty
if
\[
(
i)
E
is null,
(ii)
sup
y
∈
E
sup
t
>
0
|
∫
0
t
e
−
i
y
s
d
α
(
s
)
|
>
∞
,
(
iii)
lim
δ
↓
0
lim
sup
t
→
∞
sup
t
−
δ
⩽
s
⩽
t
|
α
(
s
)
−
α
(
t
)
|
=
0.
\begin {gathered} ({\text {i)}}\quad E\,{\text {is null,}} \hfill \\ {\text {(ii)}}\quad \sup \limits _{y \in E} \sup \limits _{t > 0} \left | {\int _0^t {{e^{ - iys}}\,d\alpha (s)} } \right | > \infty , \hfill \\ ({\text {iii)}}\quad \lim \limits _{\delta \downarrow 0} \lim \sup \limits _{t \to \infty } \sup \limits _{t - \delta \leqslant s \leqslant t} |\alpha (s) - \alpha (t)| = 0. \hfill \\ \end {gathered}
\]
This result contains Tauberian theorems for Laplace transforms, power series, and Dirichlet series.