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About Lebesgue integration...
How much set theory and measure theory should I need behind me before I'm able to make out what seems like unwieldly Lebesgue integration?
I want to apply Lebesgue integration-based theorems in Fourier analysis for functions that are not integrable by Riemann.
I want to apply Lebesgue integration-based theorems in Fourier analysis for functions that are not integrable by Riemann.
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Don't get freaked by Lebesgue integration! It's effectively a means to identify functions that are 'integrable' in a way that Riemann integration can't address. he way I remember my analysis prof describing it (when I was an undergrad) is that Lebesque worked with subsets of a function's range as opposed to the 'normal' way we work with subsets of a function's domain.
To do what you seek to do (i.e., apply Fourier-theroetic theorems to functions that are not Reimann integrable), it might do you well to identify characteristics of functions that are Lebesque integrable but not Reimann integrable, e.g., but reviewing in the book 'Counterexamples in Analysis' by Gelbaum and Almstead.
To do what you seek to do (i.e., apply Fourier-theroetic theorems to functions that are not Reimann integrable), it might do you well to identify characteristics of functions that are Lebesque integrable but not Reimann integrable, e.g., but reviewing in the book 'Counterexamples in Analysis' by Gelbaum and Almstead.
source(s):
Personal experience
Personal experience
Lebesgue integration is defined for sets which can be assigned a volume (Lebesgue measure). To understand the theory of Lebesgue measure you need to be familiar with the concepts of sigma-algebra and borel algebra (perhaps Hausdorff dimension as well)
Concerning set theory, some basic topology knowledge should be enough. (open/closed sets, compact topological spaces, topological vector spaces...)
Concerning set theory, some basic topology knowledge should be enough. (open/closed sets, compact topological spaces, topological vector spaces...)
source(s):
http://en.wikipedia.org/wiki/Lebesgue_integration
Armstrong, M.A. Basic Topology (http://www.springer.com/math/geometry/book/978-0-387-90839-7)
http://en.wikipedia.org/wiki/Lebesgue_integration
Armstrong, M.A. Basic Topology (http://www.springer.com/math/geometry/book/978-0-387-90839-7)
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I do lean towards the book reference and a more personal answer, however.