= Solution
Condition first on the count in the <random sum of independent claims>. For $N=k$, <independence> of the summands and independence from $N$ give
$$
\mathbb E[e^{uS}\mid N=k]=\prod_{j=1}^k\mathbb E[e^{uX_j}]=M_X(u)^k.
$$
The empty product is one when $k=0$. Taking <expected values> proves the <random-sum transform identity>:
$$
\boxed{M_S(u)=\sum_{k=0}^{\infty}\mathbb P(N=k)M_X(u)^k=G_N(M_X(u)).}
$$
For positive claim sizes this composition is always finite for $u\le0$, and is an identity of <Laplace transforms of nonnegative random variables> after writing $u=-s$. For positive $u$, it holds wherever the composed series is finite; no positive <exponential moment> is implied merely by positivity of the summands.
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