We consider an infinite-horizon optimal control problem with the cost functional described either by an integral over an unbounded interval (a Lebesgue integral) or by a limit of integrals (an improper Lebesgue integral). We prove some theorems on the existence of solutions to such problems. The proofs are based on appropriate lower closure theorems and some extensions of Olech’s theorem on the...

Currently published HOL formalizations of measure theory concentrate on the Lebesgue integral and they are restricted to realvalued measures. We lift this restriction by introducing the extended real numbers. We define the Borel σ-algebra for an arbitrary type forming a topological space. Then, we introduce measure spaces with extended real numbers as measure values. After defining the Lebesgue...

Summary The goal of this article is to clarify the relationship between Riemann and Lebesgue integrals. In previous [5], we constructed a one-dimensional measure. measure provides any intervals, which can be used prove well-known [6] integrals [1]. We also proved integral given that its complete As result work, bounded real valued function in Mizar system [2], [3] calculated by integral.