Abstract

In 1984, Johnson[A bounded convergence theorem for the Feynman in-tegral, J, Math. Phys, 25(1984), 1323-1326] proved a bounded convergence theorem for hte Feynman integral. This is the first stability theorem of the Feynman integral as an <TEX>$L(L_2 (\mathbb{R}^N), L_2(\mathbb{R}^{N}))$</TEX> theory. Johnson and Lapidus [Generalized Dyson series, generalized Feynman digrams, the Feynman integral and Feynmans operational calculus. Mem, Amer, Math, Soc. 62(1986), no 351] studied stability theorems for the Feynman integral as an <TEX>$L(L_2 (\mathbb{R}^N), L_2(\mathbb{R}^{N}))$</TEX> theory for the functional 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 <TEX>$L(L_1 (\mathbb{R}^N), L_{\infty}(\mathbb{R}^{N}))$</TEX>theory for the functionals which involve double integral with some Borel measures.