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Moreover, a function f defined on a bounded interval is Riemann-integrable if and only if it is bounded and the set of points where f is discontinuous has Lebesgue measure zero. An integral which is in fact a direct generalization of the Riemann integral is the Henstock–Kurzweil integral .
The fence is the section of the g(x)-sheet (i.e., the g(x) curve extended along the f(x) axis) that is bounded between the g(x)-x plane and the f(x)-sheet. The Riemann-Stieltjes integral is the area of the projection of this fence onto the f(x)-g(x) plane — in effect, its "shadow". The slope of g(x) weights the area of the projection. The ...
of a Riemann integrable function defined on a closed and bounded interval are the real numbers and , in which is called the lower limit and the upper limit. The region that is bounded can be seen as the area inside a {\displaystyle a} and b {\displaystyle b} .
Although all bounded piecewise continuous functions are Riemann-integrable on a bounded interval, subsequently more general functions were considered—particularly in the context of Fourier analysis—to which Riemann's definition does not apply, and Lebesgue formulated a different definition of integral, founded in measure theory (a subfield ...
A version holds for Fourier series as well: if is an integrable function on a bounded interval, then the Fourier coefficients ^ of tend to 0 as . This follows by extending f {\displaystyle f} by zero outside the interval, and then applying the version of the Riemann–Lebesgue lemma on the entire real line.
The Lebesgue criterion for integrability states that a bounded function is Riemann integrable if and only if the set of all discontinuities has measure zero. [5] Every countable subset of the real numbers - such as the rational numbers - has measure zero, so the above discussion shows that Thomae's function is Riemann integrable on any interval.
Since the sequence is uniformly bounded, there is a real number M such that |f n (x)| ≤ M for all x ∈ S and for all n. Define g(x) = M for all x ∈ S. Then the sequence is dominated by g. Furthermore, g is integrable since it is a constant function on a set of finite measure. Therefore, the result follows from the dominated convergence ...
The Riemann integral of a function defined over an arbitrary bounded n-dimensional set can be defined by extending that function to a function defined over a half-open rectangle whose values are zero outside the domain of the original function. Then the integral of the original function over the original domain is defined to be the integral of ...