Equivalence of integrals

Chatfield, J. A.

doi:10.1090/S0002-9939-1973-0311847-8

Equivalence of integrals
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by J. A. Chatfield

Proc. Amer. Math. Soc. 38 (1973), 279-285

DOI: https://doi.org/10.1090/S0002-9939-1973-0311847-8

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Abstract:

Suppose $R$ is the set of real numbers and $N$ is the set of nonnegative real numbers, each of $G$ and $F$ is a function from $R \times R$ to $N$. All integrals considered are of the subdivision-refinement type. This paper gives necessary and sufficient conditions for $\int _a^b {F = } \int _a^b G$. A necessary and sufficient condition for $\int _a^b {{G^2} = 0}$ is also given.

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On integration with respect to a function of bounded variation

13