B. Jessen

The theory of integration in a space of an infinite number of dimensions

Introduction. The torus-space in an infinite number of dimensions. Limit points. Covering theorems. Continuous and semi-continuous functions. The Lebesgue measure. The construction of nets. The transferring principle. The Jordan measure. The definite and indefinite integrals. The Riemann integral. An important lemma. Application Of Fubinis theorem. Infinitely multiple integrals. Representation of a function as the limit of an integral. Strong convergence. Majorised convergence. Fourier series. A special orthogonal system. A case of Birkhoffs ergodic theorem. Application to almost periodic functions. 0rthogonal series whose coefficients are analytic functions. General exponential series. Analytic almost periodic functions. Distribution functions.