Specifically, as finer partitions of a given interval are considered, their mesh approaches zero and the Riemann sum based on a given partition approaches the Riemann integral.
42.
The narrow definition of the Riemann integral also does not cover the function 1 / \ sqrt { x } on the interval [ 0, 1 ].
43.
For example, the Dirichlet function, which is 0 where its argument is irrational and 1 otherwise, has a Lebesgue integral, but does not have a Riemann integral.
44.
This book is practically self-contained apart form the very elementary stuff ( things like definition of limits, derivative, Riemann integral, uniform convergence etc . etc . ).
45.
:If you want to get pedantic, then the Riemann integral does not exist, but the improper integral ( and the Lebesgue integral, IIRC ) evaluates to + infinity.
46.
Much stronger theorems in this respect, which require not much more than pointwise convergence, can be obtained if one abandons the Riemann integral and uses the Lebesgue integral instead.
47.
An important generalization is the Lebesgue & ndash; Stieltjes integral which generalizes the Riemann Stieltjes integral in a way analogous to how the Lebesgue integral generalizes the Riemann integral.
48.
Lebesgue integration has the property that every function defined over a bounded interval with a Riemann integral also has a Lebesgue integral, and for those functions the two integrals agree.
49.
*Since step functions are integrable and the integrability and the value of a Riemann integral are compatible with uniform limits, the regulated integral is a special case of the Riemann integral.
50.
If one has a continuous function f : R \ to R such that the improper Riemann integral over R is infinite, then is the Lebesgue integral over R also infinite?
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