The paper considers the Feynman-Kac penalization problem, i.e., (i) does there exist a probability measure \(\tilde{\operatorname{P}}_x\) such that \[ \lim_{t\to\infty}\frac{\operatorname{E}_x [e^{A_t}S]}{\operatorname{E}_x[e^{A_t}]} =\int Sd\tilde{\operatorname{P}}_x, \] for any bounded \(S\), every \(x\in R^n\), \(s\geq 0\), where \(A_t\) is an additive functional of a symmetric \(\alpha\)-stable process \(X\) with \(0<\alpha<2\), and (ii) does there exists a martingale \(M\) by which \(\tilde{\operatorname{P}}\) is defined (that is, \(d\tilde{\operatorname{P}}_x = M_sd\operatorname{P}_x\)). The author solves this problem for additive functionals with jumps, that is \(A_t\) is a positive continuous additive functional with Green-tight Revuz measure to which symmetric jumps are added. Examples of jumping functions are given.