The condition of our skin often changes with the seasons. In fact, it changes throughout our lives, responding to our health, our skincare, our lifestyle and our surroundings. Our skin often deteriorates during cold weather, and this book explores why this happens and what we can do about it. This book connects our skin with our lifestyle, our wellbeing and our environment, and is relevant to everyone with an interest in these areas.
The problem as to whether or not there exists a lifting of the M't/. 1 space ) corresponding to the real line and Lebesgue measure on it was first raised by A. Haar. It was solved in a paper published in 1931 [102] by 1. von Neumann, who established the existence of a lifting in this case. In subsequent papers J. von Neumann and M. H. Stone [105], and later on 1. Dieudonne [22], discussed various algebraic aspects and generalizations of the problem. Attemps to solve the problem as to whether or not there exists a lifting for an arbitrary M't/. space were unsuccessful for a long time, although the problem had significant connections with other branches of mathematics. Finally, in a paper published in 1958 [88], D. Maharam established, by a delicate argument, that a lifting of M't/. always exists (for an arbi trary space of a-finite mass). D. Maharam proved first the existence of a lifting of the M't/. space corresponding to a product X = TI {ai,b,} ieI and a product measure J.1= Q9 J.1i' with J.1i{a;}=J.1i{b,}=! for all iE/. ,eI Then, she reduced the general case to this one, via an isomorphism theorem concerning homogeneous measure algebras [87], [88]. A different and more direct proof of the existence of a lifting was subsequently given by the authors in [65]' A variant of this proof is presented in chapter 4.
Examining non-technological innovations for environmentally and socially-friendly transport, this book provides the reader with a better understanding of this often overlooked topic. It features four illustrative case studies, and presents a concise review of the core transport modes (road, rail and marine transport). Transport companies are compelled to innovate due to economic and environmental pressures, and the aim of these innovations is to improve fuel efficiency and ultimately to transform energy use in the transport sector. Whilst many of these innovations are technological, they can conversely be non-technological in nature. This book is intended for students and researchers interested in economics, environmental economics and economics of innovation. It also offers a useful resource to industry professionals interested in ecology and transport.
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