Jae Myung Park

The Denjoy extension of the Riemann and McShane integrals

In this paper we study the Denjoy-Riemann and Denjoy-McShane integrals of functions mapping an interval [a, b] into a Banach space X. It is shown that a Denjoy-Bochner integrable function on [a, b] is Denjoy-Riemann integrable on [a, b], that a Denjoy-Riemann integrable function on [a, b] is Denjoy-McShane integrable on [a, b] and that a Denjoy-McShane integrable function on [a, b] is Denjoy-Pettis integrable on [a, b]. In addition, it is shown that for spaces that do not contain a copy of c0, a measurable Denjoy-McShane integrable function on [a, b] is McShane integrable on some subinterval of [a, b]. Some examples of functions that are integrable in one sense but not another are included.

THE STRONG PERRON INTEGRAL IN THE n-DIMENSIONAL SPACE ℝ n

In this paper, we introduce the SP-integral and the SPfi-integral deflned on an interval in the n-dimensional Euclidean space R n . We also investigate the relationship between these two integrals. It is well known (3) that the Perron integral deflned on an interval of the real line R by major and minor functions which are not assumed to be continuous is equivalent to the one deflned by continuous major and minor functions and that the strong Perron integral deflned on an interval of R by strong major and minor functions is equivalent to the McShane integral. In this paper, we introduce Perron-type integrals deflned on an inter- val of the n-dimensional Euclidean spaceR n using the strong major and minor functions, and investigate the relationship between these integrals. We shall call it the strong Perron integral, or brie∞y SP-integral. For a subset E of the n-dimensional Euclidean spaceR n , the Lebesgue measure of E is denoted by jEj. For a point x = (x1;x2;¢¢¢ ;xn) 2R n , the norm of x is kxk = max1•in jxij and the --neighborhood U(x;-) of x is an open cube centered at x with sides equal to 2-. For an interval I = (a1;b1)£(a2;b2)£¢¢¢(an;bn) ofR n with ai fi(fi 2 (0;1)), then the interval I is said to be fi-regular.