In 1984, Johnson (A bounded convergence theorem for the Feynman integral, J. Math. Phys. 25 (1984), 1323-1326 proved a bounded convergence theorem for the Feynman integral. This is the first stability theorem of the Feynman integral as an L(L₂(R^N), L₂(R^N)) theory. Johnson and Lapidus [Generalized Dyson series, generalized Feynman digrams, the Feynman integral and Feynman`s operational calculus. Mem. Amer. Math. Soc. 62 (1986), no. 351 studied stability theorems for the Feynman integral as an L(L₂(R^N), L₂(R^N)) theory for the functionals with arbitrary Borel measure. These papers treat functionals which involve only a single integral. In this paper, we obtain the stability theorems for the Feynman integral as an L(L₁(R), L_∞(R)) theory for the functionals which involve double integral with some Borel measures.