Product Pre-Measure

Summary In this article we formalize in Mizar [5] product pre-measure on product sets of measurable sets. Although there are some approaches to construct product measure [22], [6], [9], [21], [25], we start it from σ-measure because existence of σ-measure on any semialgebras has been proved in [15]. In this approach, we use some theorems for integrals.

[1]  Yasunari Shidama,et al.  The Lebesgue Monotone Convergence Theorem , 2008, Formaliz. Math..

[2]  Wojciech A. Trybulec Non-contiguous Substrings and One-to-one Finite Sequences , 1990 .

[3]  Noboru Endou The Measurability of Extended Real Valued Functions , 2000 .

[4]  Grzegorz Bancerek,et al.  Curried and Uncurried Functions , 1990 .

[5]  Czeslaw Bylinski Functions and Their Basic Properties , 2004 .

[6]  Czeslaw Bylinski,et al.  Basic Functions and Operations on Functions , 1989 .

[7]  Józef Bia Series of Positive Real Numbers . Measure Theory , 1990 .

[8]  Grzegorz Bancerek,et al.  Towards the Construction of a Model of Mizar Concepts , 2008, Formaliz. Math..

[9]  Andrzej Trybulec,et al.  Binary Operations Applied to Functions , 1990 .

[10]  Czeslaw Bylinski Functions from a Set to a Set , 2004 .

[11]  M. Rao Measure Theory and Integration , 2018 .

[12]  Beata Perkowska,et al.  Functional Sequence from a Domain to a Domain , 1992 .

[13]  Yasunari Shidama,et al.  Inferior Limit, Superior Limit and Convergence of Sequences of Extended Real Numbers , 2007 .

[14]  Yasunari Shidama,et al.  Hopf Extension Theorem of Measure , 2009, Formaliz. Math..

[15]  J. K. Hunter,et al.  Measure Theory , 2007 .

[16]  Noboru Endou,et al.  Construction of Measure from Semialgebra of Sets1 , 2015, Formaliz. Math..

[17]  Yasunari Shidama,et al.  Integral of Measurable Function1 , 2006 .

[18]  Adam Naumowicz,et al.  Mizar: State-of-the-art and Beyond , 2015, CICM.

[19]  Gerald B. Folland,et al.  Real Analysis: Modern Techniques and Their Applications , 1984 .

[20]  Edmund Woronowicz Relations and Their Basic Properties , 2004 .

[21]  Józef Bia las The σ-additive Measure Theory , 1990 .

[22]  G. Bancerek The Fundamental Properties of Natural Numbers , 1990 .

[23]  Artur Korni,et al.  Some Basic Properties of Many Sorted Sets , 1996 .

[24]  Andrzej Ne ' dzusiak Fields and Probability , 1990 .

[25]  Noboru Endou Definitions and Basic Properties of Measurable Functions , 2004 .