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Examples
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Now that we have reviewed the foundations of measure theory, let us now put it to work to set up the basic theory of one of the fundamental families of spaces (also known as Lebesgue spaces).
What's new 2009
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In order to analyse how solutions behave in each of these regimes rigorously, one usually works with a variety of function spaces (such as Lebesgue spaces
What's new Terence Tao 2010
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In order to analyse how solutions behave in each of these regimes rigorously, one usually works with a variety of function spaces (such as Lebesgue spaces
What's new Terence Tao 2010
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To construct a Lebesgue integral, begin instead by partitioning the y-axis.
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The Lebesgue integral is at the heart of measure theory and probability theory, and I had long thought of it as elegant but difficult to visualize.
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This construction leads to the Lebesgue integral and is beautifully illustrated by the Demonstration “Riemann versus Lebesgue” by Francisco J. Freniche.
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It took the better part of a yearlong course in analysis for me to appreciate the connection between this new, more general Lebesgue integral and my old friend the Riemann integral.
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There are, however, situations in which the Lebesgue approximations converge and the Riemann approximations do not.
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The Lebesgue integral turns out to be a generalization of the Riemann integral in the sense that if the Riemann approximations converge, so do the Lebesgue approximations and to the same value.
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There are many null sets, for example the set of integers (and all subsets thereof) considering Lebesgue measure.
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