Let $Γ$ be an LCA group and $(μ_n)$ be a sequence of bounded regular Borel measures on $Γ$ tending to a measure $μ_0$. Let $G$ be the dual group of $Γ$, $S$ be a non-empty subset of $G \setminus \{ 0 \}$, and $[{\mathcal T}(S)]_{μ_n,p}$ the subspace of $L^p(μ_n)$, $p \in (0,\infty)$, spanned by the characters of $Γ$ which are generated by the elements of $S$. The limit behaviour of the sequence of metric projections of the function $1$ onto $[{\mathcal T}(S)]_{μ_n,p}$ as well as of the sequence of the corresponding approximation errors are studied. The results are applied to obtain stability theorems for prediction of weakly stationary or harmonizable symmetric $p$-stable stochastic processes. Along with the general problem the particular cases of linear interpolation or extrapolation as well as of a finite or periodic observation set are studied in detail and compared to each other.