Criteria for $L^1$ convergence looking at Laplace transforms Let $(X_n)_{n \geq 0}$ be a sequence of integrable ($\mathbb{E} |X_n| < \infty$) random variables  and denote by $l_n(t)$ the Laplace transforms of $X_n$. Similarly, let $X$ be a r.v. and $l(t)$ its Laplace transform.
It is known (Levy continuity theorem for Laplace transforms) that $l_n(t) \to l(t)$ for every $t>0$ iff $X_n \Longrightarrow X$ (where with the notation $X_n \Longrightarrow X$ I mean that $X_n$ converges weakly/in distribution to $X$).
Denote by $g'$ the first derivative of $g$. I was wondering whether additional hypotheses such $l_n'(t) \to l'(t)$ for every $t>0$ (or maybe simply $l'_n(0) \to l(0)'$) imply that $X_n \stackrel{L^1}{\to} X$.
 A: It's an interesting question to think about. I came across it from the following problem:
pointwise convergence of Laplace transform implies weak convergence of probability measures $\mathbb{P}_n \xrightarrow{w} \mathbb{P}$. Is there anyproperty on Laplace transform so probabilities converge in total variation norm?
$$
  \sup\limits_{A\in \mathbb{B}(R)}|\mathbb{P}_n(A) - \mathbb{P}(A)| \rightarrow 0
$$
There is an interesting result about characteristic functions on this topic (look for Van der Vaart, Asymptotic Statistics, Cambridge Univ. Press):
Theorem: Consider $Y_1,\dots, Y_n$ -- i.i.d r.v. with $\mathbb{E}Y = 0, \, \mathbb{E}Y^2 < +\infty$. And let characteristic function $\phi(t)$ satisfies the following
$$
 \int |\phi(t)|^\nu dt < \infty, \text{ for some } \nu \geq 1,
$$
then 
$$
\dfrac{Y_1 + \dots + Y_n}{\sqrt{n}} \xrightarrow{TV} \mathcal{N}(0,1)
$$
Remark: Total variation -- is equal to $L^1$ distance between densities, so this property implies $L^1$ -- convergence. Proof is following from Sheffe corollary
Corollary (Sheffe): Consider $\{p_n\}$, p -- densities wit respect to dominating measure $\mu$. If $p_n\rightarrow p$ $\mu$ -- almost everywhere then random values $X_n \xrightarrow{TV} X$, where $X_n$ have density $p_n$ correspondingly and $X$ has density $p$.
Conclusion:
I still think on it, but if I want to use Sheffe corollary to investigate the problem for Laplace, I have to study the inverse Laplace transform. If through it you can show almost everywhere convergence of densities -- then you will have an $L^1$ type of convergence.
$\textbf{Another solution}:$ Thanks to nice talk with professor Yuri Davydov from Paris 6 Univ. there is a link to the nice property of Laplace-Stiltjes transform related to strong convergence (convergence in $L^1$ for densities):
Kovács, Mihály, and Frank Neubrander. "On the inverse Laplace-Stieltjes transform of A-stable rational functions." New Zealand J. Math 36 (2007): 41-56. 
