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We consider the analogue of Wiener measure on the space of all real continuous functions, and then establish the measure-valued measure and the operator-valued measure on our space and we investigate their important properties, for example, the integration formula, the rotation theorem, the Bartle integral, the Bochner integral and the Dobrakov integral, the simple formula for conditional expectation, the measure-valued Feynman-Kac formula and Volterra integral equation and integral transform.

Abstract. In this article, we consider a complex-valued and a measure-valued measure on C [0, t], the space of all real-valued continuous functions on [0, t]. Using these concepts, we establish the measure-valued Feynman-Kac formula and we prove that this formula satisfies a Volterra integral equation. The work here is patterned to some extent on earlier works by Kluvanek in 1983 and by Lapidus in 1987, but the present setting requires a number of new concepts and results.

Abstract. In this note, we introduce the definition of the generalized analogue of Wiener measure on the space C [a, b] of all realvalued continuous functions on the closed interval [a, b], give several examples of it and investigate some important properties of it -the Fernique theorem and the existence theorem of scale-invariant measurable subsets on C [a, b]. PreliminariesIn 1923, Wiener proved the existence theorem of the meaningly measure on the space C 0 [a, b], the space of all real-valued continuous functions on a closed bounded interval [a, b] which vanish at a, the so-called Wiener space in [9]. This is based on the properties of Brownian motion of a single small particle. In 2002, the author and Dr. Im presented the definition and properties of analogue of Wiener measure on the space C [a, b], the space of all real-valued continuous functions on [a, b] in [3]. This is the theory of many particles, moving along the law of Brownian motion.In this note, we introduce the definition of the generalized analogue of Wiener measure space, which is more generalized concept of Wiener measure space and we give several examples. Furthermore, we investigate important theorems -the Fernique theorem on C [a, b] and the existence theorem of scale-invariant measurable subsets on C [a, b].In this section, we give some notations, definitions and facts which are needed to understand the subsequent sections.

Abstract. In this note, we prove the translation theorem for the generalized analogue of Wiener measure space and we show some properties of the generalized analogue of Wiener measure from it. PreliminariesWiener proved the existence theorem of the measure m w , based on the properties of Brownian motion, on the space . This is the generalization of the concrete Wiener measure space, including both concepts of the