Measure Theory and the Central Limit Theorem
نویسنده
چکیده
It has long been known that the standard Riemann integral, encountered first as one of the most important concepts of calculus, has many shortcomings, and can only be defined over the limited class of sets which have boundaries of Jordan content zero. In this paper I present the measure theory necessary to develop the more robust Lebesgue integral. This is interesting in its own right, and although we dwell little on the consequences of our definition, any calculus student who is familiar with the properties of the Riemann integral will recognize its tremendous advantages. But we will push further: measure theory and the Lebesgue integral provide us with the ideal background to develop a rigorous theory of probability. To this end, we introduce random variables and develop the theory of distribution functions. This paper culminates in the proof of the Central Limit Theorem (CLT), which explains the ubiquitous nature of the normal distribution. This is a deep and fascinating result, but relatively straightforward once one has provided the correct machinery. Indeed, we do most of the hard work studying measure theory. This paper does not provide the most intuitive approach towards the CLT, but it has the advantage of providing an introduction to rigorous probability that may serve as a starting point for future study. We have tried to provide the most expedient treatment of the material possible without sacrificing the intuition or the depth of the theoretical background.
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تاریخ انتشار 2011