The Riemann Integral - Revision history

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Definition

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Lila: /* {{anchor|Riemann-integrable}} Riemann integral */

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{{anchor|Riemann-integrable}} Riemann integral

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Licensing

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 Some calculus books do not use general tagged partitions, but limit themselves to specific types of tagged partitions. If the type of partition is limited too much, some non-integrable functions may appear to be integrable.
-One popular restriction is the use of "left-hand" and "right-hand" Riemann sums. In a left-hand Riemann sum, {{math|''t<sub>i</sub>'' = ''x<sub>i</sub>''}} for all {{mvar|i}}, and in a right-hand Riemann sum, {{math|''t<sub>i</sub>'' = ''x''<sub>''i'' + 1</sub>}} for all {{mvar|i}}. Alone this restriction does not impose a problem: we can refine any partition in a way that makes it a left-hand or right-hand sum by subdividing it at each {{mvar|t<sub>i</sub>}}. In more formal language, the set of all left-hand Riemann sums and the set of all right-hand Riemann sums is cofinal in the set of all tagged partitions.
+One popular restriction is the use of "left-hand" and "right-hand" Riemann sums. In a left-hand Riemann sum, ''t<sub>i</sub>'' = ''x<sub>i</sub>'' for all {{mvar|i}}, and in a right-hand Riemann sum, ''t<sub>i</sub>'' = ''x''<sub>''i'' + 1</sub> for all {{mvar|i}}. Alone this restriction does not impose a problem: we can refine any partition in a way that makes it a left-hand or right-hand sum by subdividing it at each {{mvar|t<sub>i</sub>}}. In more formal language, the set of all left-hand Riemann sums and the set of all right-hand Riemann sums is cofinal in the set of all tagged partitions.
 Another popular restriction is the use of regular subdivisions of an interval. For example, the {{mvar|n}}th regular subdivision of {{math|[0, 1]}} consists of the intervals

@@ Line 235: / Line 235: @@
 :<math>\lim_{a\to\infty} \int_{-a}^a f(x)\,dx.</math>
-For example, consider the sign function {{math|''f''(''x'') = sgn(''x'')}} which is 0 at {{math|''x'' = 0}}, 1 for {{math|''x'' > 0}}, and −1 for {{math|''x'' < 0}}. By symmetry,
+For example, consider the sign function ''f''(''x'') = sgn(''x'') which is 0 at ''x'' = 0, 1 for {{math|''x'' > 0}}, and −1 for {{math|''x'' < 0}}. By symmetry,
 :<math>\int_{-a}^a f(x)\,dx = 0</math>

@@ Line 33: / Line 33: @@
 A '''tagged partition''' {{math|''P''(''x'', ''t'')}} of an interval {{math|[''a'', ''b'']}} is a partition together with a finite sequence of numbers {{math|''t''<sub>0</sub>, ..., ''t''<sub>''n'' − 1</sub>}} subject to the conditions that for each {{mvar|i}}, {{math|''t<sub>i</sub>'' ∈ [''x<sub>i</sub>'', ''x''<sub>''i'' + 1</sub>]}}. In other words, it is a partition together with a distinguished point of every sub-interval. The mesh of a tagged partition is the same as that of an ordinary partition.
-Suppose that two partitions {{math|''P''(''x'', ''t'')}} and {{math|''Q''(''y'', ''s'')}} are both partitions of the interval {{math|[''a'', ''b'']}}. We say that {{math|''Q''(''y'', ''s'')}} is a '''refinement''' of {{math|''P''(''x'', ''t'')}} if for each integer {{mvar|i}}, with {{math|''i'' ∈ [0, ''n'']}}, there exists an integer {{math|''r''(''i'')}} such that {{math|''x<sub>i</sub>'' = ''y''<sub>''r''(''i'')</sub>}} and such that {{math|''t<sub>i</sub>'' = ''s<sub>j</sub>''}} for some {{mvar|j}} with {{math|''j'' ∈ [''r''(''i''), ''r''(''i'' + 1))}}. Said more simply, a refinement of a tagged partition breaks up some of the sub-intervals and adds tags to the partition where necessary, thus it "refines" the accuracy of the partition.
+Suppose that two partitions {{math|''P''(''x'', ''t'')}} and {{math|''Q''(''y'', ''s'')}} are both partitions of the interval {{math|[''a'', ''b'']}}. We say that {{math|''Q''(''y'', ''s'')}} is a '''refinement''' of {{math|''P''(''x'', ''t'')}} if for each integer {{mvar|i}}, with {{math|''i'' ∈ [0, ''n'']}}, there exists an integer {{math|''r''(''i'')}} such that ''x<sub>i</sub>'' = ''y''<sub>''r''(''i'')</sub> and such that ''t<sub>i</sub>'' = ''s<sub>j</sub>'' for some {{mvar|j}} with {{math|''j'' ∈ [''r''(''i''), ''r''(''i'' + 1))}}. Said more simply, a refinement of a tagged partition breaks up some of the sub-intervals and adds tags to the partition where necessary, thus it "refines" the accuracy of the partition.
 We can turn the set of all tagged partitions into a directed set  by saying that one tagged partition is greater than or equal to another if the former is a refinement of the latter.

@@ Line 198: / Line 198: @@
 Thus the partition divides {{math|[''a'', ''b'']}} to two kinds of intervals:
-*Intervals of the latter kind (themselves subintervals of some {{math|''J''(''ε''<sub>1</sub>)<sub>''i''</sub>}}). In each of these, {{mvar|f}} oscillates by less than {{math|''ε''<sub>1</sub>}}. Since the total length of these is not larger than {{math|''b'' − ''a''}}, they together contribute at most {{math|1=''ε''{{su|b=1|p=∗}}(''b'' − ''a'') = ''ε''/2}} to the difference between the upper and lower sums of the partition.
+*Intervals of the latter kind (themselves subintervals of some {{math|''J''(''ε''<sub>1</sub>)<sub>''i''</sub>}}). In each of these, {{mvar|f}} oscillates by less than {{math|''ε''<sub>1</sub>}}. Since the total length of these is not larger than {{math|''b'' − ''a''}}, they together contribute at most {{math|1=''ε''<sub>1</sub><sup>*</sup>(''b'' − ''a'') = ''ε''/2}} to the difference between the upper and lower sums of the partition.
-*The intervals {{math|{''I''(''ε'')<sub>''i''</sub>}|}}. These have total length smaller than {{math|''ε''<sub>2</sub>}}, and {{mvar|f}} oscillates on them by no more than {{math|''M'' − ''m''}}. Thus together they contribute less than {{math|1=''ε''{{su|b=2|p=∗}}(''M'' − ''m'') = ''ε''/2}} to the difference between the upper and lower sums of the partition.
+*The intervals {{math|{''I''(''ε'')<sub>''i''</sub>}|}}. These have total length smaller than {{math|''ε''<sub>2</sub>}}, and {{mvar|f}} oscillates on them by no more than {{math|''M'' − ''m''}}. Thus together they contribute less than {{math|1=''ε''<sub>2</sub><sup>*</sup>(''M'' − ''m'') = ''ε''/2}} to the difference between the upper and lower sums of the partition.
 In total, the difference between the upper and lower sums of the partition is smaller than {{mvar|ε}}, as required.

@@ Line 106: / Line 106: @@
 :<math>x_{i+1}-y_{j+1} < \delta < \frac{\varepsilon}{2r(m-1)},</math>
-It follows that, for some (indeed any) {{math|''t''{{su|b=''i''|p=*}} ∈ [''y''<sub>''j'' + 1</sub>, ''x''<sub>''i'' + 1</sub>]}},
+It follows that, for some (indeed any) {{math|''t''<sub>i</sub><sup>*</sup> ∈ [''y''<sub>''j'' + 1</sub>, ''x''<sub>''i'' + 1</sub>]}},
 :<math>\left|f\left(t_i\right)-f\left(t_i^*\right)\right|\left(x_{i+1}-y_{j+1}\right) < \frac{\varepsilon}{2(m-1)}.</math>

@@ Line 53: / Line 53: @@
 If {{mvar|f}} is continuous, then the lower and upper Darboux sums for an untagged partition are equal to the Riemann sum for that partition, where the tags are chosen to be the minimum or maximum (respectively) of {{mvar|f}} on each subinterval. (When {{mvar|f}} is discontinuous on a subinterval, there may not be a tag that achieves the infimum or supremum on that subinterval.) The Darboux integral, which is similar to the Riemann integral but based on Darboux sums, is equivalent to the Riemann integral.
-=== {{anchor|Riemann-integrable}} Riemann integral ===
+===Riemann integral ===
 Loosely speaking, the Riemann integral is the limit of the Riemann sums of a function as the partitions get finer. If the limit exists then the function is said to be '''integrable''' (or more specifically '''Riemann-integrable'''). The Riemann sum can be made as close as desired to the Riemann integral by making the partition fine enough.

@@ Line 33: / Line 33: @@
 A '''tagged partition''' {{math|''P''(''x'', ''t'')}} of an interval {{math|[''a'', ''b'']}} is a partition together with a finite sequence of numbers {{math|''t''<sub>0</sub>, ..., ''t''<sub>''n'' − 1</sub>}} subject to the conditions that for each {{mvar|i}}, {{math|''t<sub>i</sub>'' ∈ [''x<sub>i</sub>'', ''x''<sub>''i'' + 1</sub>]}}. In other words, it is a partition together with a distinguished point of every sub-interval. The mesh of a tagged partition is the same as that of an ordinary partition.
-Suppose that two partitions {{math|''P''(''x'', ''t'')}} and {{math|''Q''(''y'', ''s'')}} are both partitions of the interval {{math|[''a'', ''b'']}}. We say that {{math|''Q''(''y'', ''s'')}} is a '''refinement''' of {{math|''P''(''x'', ''t'')}} if for each integer {{mvar|i}}, with {{math|''i'' ∈ [0, ''n'']}}, there exists an integer {{math|''r''(''i'')}} such that {{math|''x<sub>i</sub>'' {{=}} ''y''<sub>''r''(''i'')</sub>}} and such that {{math|''t<sub>i</sub>'' {{=}} ''s<sub>j</sub>''}} for some {{mvar|j}} with {{math|''j'' ∈ [''r''(''i''), ''r''(''i'' + 1))}}. Said more simply, a refinement of a tagged partition breaks up some of the sub-intervals and adds tags to the partition where necessary, thus it "refines" the accuracy of the partition.
+Suppose that two partitions {{math|''P''(''x'', ''t'')}} and {{math|''Q''(''y'', ''s'')}} are both partitions of the interval {{math|[''a'', ''b'']}}. We say that {{math|''Q''(''y'', ''s'')}} is a '''refinement''' of {{math|''P''(''x'', ''t'')}} if for each integer {{mvar|i}}, with {{math|''i'' ∈ [0, ''n'']}}, there exists an integer {{math|''r''(''i'')}} such that {{math|''x<sub>i</sub>'' = ''y''<sub>''r''(''i'')</sub>}} and such that {{math|''t<sub>i</sub>'' = ''s<sub>j</sub>''}} for some {{mvar|j}} with {{math|''j'' ∈ [''r''(''i''), ''r''(''i'' + 1))}}. Said more simply, a refinement of a tagged partition breaks up some of the sub-intervals and adds tags to the partition where necessary, thus it "refines" the accuracy of the partition.
 We can turn the set of all tagged partitions into a directed set  by saying that one tagged partition is greater than or equal to another if the former is a refinement of the latter.
@@ Line 54: / Line 54: @@
 === {{anchor|Riemann-integrable}} Riemann integral ===
-Loosely speaking, the Riemann integral is the limit of the Riemann sums of a function as the partitions get finer. If the limit exists then the function is said to be '''integrable''' (or more specifically '''Riemann-integrable'''). The Riemann sum can be made as close as desired to the Riemann integral by making the partition fine enough.<ref>{{Cite book|last=Taylor|first=Michael E.|author-link=Michael E. Taylor|title=Measure Theory and Integration| publisher=American Mathematical Society|year=2006|isbn=9780821872468|page=1|url=https://books.google.com/books?id=P_zJA-E5oe4C&pg=PA1}}</ref>
+Loosely speaking, the Riemann integral is the limit of the Riemann sums of a function as the partitions get finer. If the limit exists then the function is said to be '''integrable''' (or more specifically '''Riemann-integrable'''). The Riemann sum can be made as close as desired to the Riemann integral by making the partition fine enough.
 One important requirement is that the mesh of the partitions must become smaller and smaller, so that in the limit, it is zero. If this were not so, then we would not be getting a good approximation to the function on certain subintervals. In fact, this is enough to define an integral. To be specific, we say that the Riemann integral of {{mvar|f}} equals {{mvar|s}} if the following condition holds:
@@ Line 76: / Line 76: @@
 :<math> r = 2\sup_{x \in [a, b]} |f(x)|.</math>
-If {{math|''r'' {{=}} 0}}, then {{mvar|f}} is the zero function, which is clearly both Darboux and Riemann integrable with integral zero. Therefore, we will assume that {{math|''r'' > 0}}. If {{math|''m'' > 1}}, then we choose {{mvar|δ}} such that
+If {{math|''r'' = 0}}, then {{mvar|f}} is the zero function, which is clearly both Darboux and Riemann integrable with integral zero. Therefore, we will assume that {{math|''r'' > 0}}. If {{math|''m'' > 1}}, then we choose {{mvar|δ}} such that
 :<math>\delta < \min \left \{\frac{\varepsilon}{2r(m-1)}, \left(y_1 - y_0\right), \left(y_2 - y_1\right), \cdots, \left(y_m - y_{m-1}\right) \right \}</math>
-If {{math|''m'' {{=}} 1}}, then we choose {{mvar|δ}} to be less than one. Choose a tagged partition {{math|''x''<sub>0</sub>, ..., ''x<sub>n</sub>''}} and {{math|''t''<sub>0</sub>, ..., ''t''<sub>''n'' − 1</sub>}} with mesh smaller than {{mvar|δ}}. We must show that the Riemann sum is within {{mvar|ε}} of {{mvar|s}}.
+If {{math|''m'' = 1}}, then we choose {{mvar|δ}} to be less than one. Choose a tagged partition {{math|''x''<sub>0</sub>, ..., ''x<sub>n</sub>''}} and {{math|''t''<sub>0</sub>, ..., ''t''<sub>''n'' − 1</sub>}} with mesh smaller than {{mvar|δ}}. We must show that the Riemann sum is within {{mvar|ε}} of {{mvar|s}}.
 To see this, choose an interval {{math|[''x<sub>i</sub>'', ''x''<sub>''i'' + 1</sub>]}}. If this interval is contained within some {{math|[''y<sub>j</sub>'', ''y''<sub>''j'' + 1</sub>]}}, then
@@ Line 86: / Line 86: @@
 :<math> m_j < f(t_i) < M_j</math>
-where {{mvar|m<sub>j</sub>}} and {{mvar|M<sub>j</sub>}} are respectively, the infimum and the supremum of ''f'' on {{math|[''y<sub>j</sub>'', ''y''<sub>''j'' + 1</sub>]}}. If all intervals had this property, then this would conclude the proof, because each term in the Riemann sum would be bounded by a corresponding term in the Darboux sums, and we chose the Darboux sums to be near {{mvar|s}}. This is the case when {{math|''m'' {{=}} 1}}, so the proof is finished in that case.
+where {{mvar|m<sub>j</sub>}} and {{mvar|M<sub>j</sub>}} are respectively, the infimum and the supremum of ''f'' on {{math|[''y<sub>j</sub>'', ''y''<sub>''j'' + 1</sub>]}}. If all intervals had this property, then this would conclude the proof, because each term in the Riemann sum would be bounded by a corresponding term in the Darboux sums, and we chose the Darboux sums to be near {{mvar|s}}. This is the case when {{math|''m'' = 1}}, so the proof is finished in that case.
 Therefore, we may assume that {{math|''m'' > 1}}. In this case, it is possible that one of the {{math|[''x<sub>i</sub>'', ''x''<sub>''i'' + 1</sub>]}} is not contained in any {{math|[''y<sub>j</sub>'', ''y''<sub>''j'' + 1</sub>]}}. Instead, it may stretch across two of the intervals determined by {{math|''y''<sub>0</sub>, ..., ''y<sub>m</sub>''}}. (It cannot meet three intervals because {{mvar|δ}} is assumed to be smaller than the length of any one interval.) In symbols, it may happen that
@@ Line 141: / Line 141: @@
 Some calculus books do not use general tagged partitions, but limit themselves to specific types of tagged partitions. If the type of partition is limited too much, some non-integrable functions may appear to be integrable.
-One popular restriction is the use of "left-hand" and "right-hand" Riemann sums. In a left-hand Riemann sum, {{math|''t<sub>i</sub>'' {{=}} ''x<sub>i</sub>''}} for all {{mvar|i}}, and in a right-hand Riemann sum, {{math|''t<sub>i</sub>'' {{=}} ''x''<sub>''i'' + 1</sub>}} for all {{mvar|i}}. Alone this restriction does not impose a problem: we can refine any partition in a way that makes it a left-hand or right-hand sum by subdividing it at each {{mvar|t<sub>i</sub>}}. In more formal language, the set of all left-hand Riemann sums and the set of all right-hand Riemann sums is cofinal in the set of all tagged partitions.
+One popular restriction is the use of "left-hand" and "right-hand" Riemann sums. In a left-hand Riemann sum, {{math|''t<sub>i</sub>'' = ''x<sub>i</sub>''}} for all {{mvar|i}}, and in a right-hand Riemann sum, {{math|''t<sub>i</sub>'' = ''x''<sub>''i'' + 1</sub>}} for all {{mvar|i}}. Alone this restriction does not impose a problem: we can refine any partition in a way that makes it a left-hand or right-hand sum by subdividing it at each {{mvar|t<sub>i</sub>}}. In more formal language, the set of all left-hand Riemann sums and the set of all right-hand Riemann sums is cofinal in the set of all tagged partitions.
 Another popular restriction is the use of regular subdivisions of an interval. For example, the {{mvar|n}}th regular subdivision of {{math|[0, 1]}} consists of the intervals
@@ Line 176: / Line 176: @@
 |The proof is easiest using the Darboux integral definition of integrability (formally, the Riemann condition for integrability) – a function is Riemann integrable if and only if the upper and lower sums can be made arbitrarily close by choosing an appropriate partition.
-One direction can be proven using the oscillation definition of continuity:<ref>[http://unapologetic.wordpress.com/2009/12/15/lebesgues-condition/ Lebesgue’s Condition], John Armstrong, December 15, 2009, The Unapologetic Mathematician</ref> For every positive {{mvar|ε}}, Let {{math|''X''<sub>''ε''</sub>}} be the set of points in {{math|[''a'', ''b'']}} with oscillation of at least {{mvar|ε}}. Since every point where {{mvar|f}} is discontinuous has a positive oscillation and vice versa, the set of points in {{math|[''a'', ''b'']}}, where {{mvar|f}} is discontinuous is equal to the union over {{math|{''X''<sub>1/''n''</sub>}|}} for all natural numbers {{mvar|n}}.
+One direction can be proven using the oscillation definition of continuity: For every positive {{mvar|ε}}, Let {{math|''X''<sub>''ε''</sub>}} be the set of points in {{math|[''a'', ''b'']}} with oscillation of at least {{mvar|ε}}. Since every point where {{mvar|f}} is discontinuous has a positive oscillation and vice versa, the set of points in {{math|[''a'', ''b'']}}, where {{mvar|f}} is discontinuous is equal to the union over {{math|{''X''<sub>1/''n''</sub>}|}} for all natural numbers {{mvar|n}}.
 If this set does not have a zero Lebesgue measure, then by countable additivity of the measure there is at least one such {{mvar|n}} so that {{math|''X''<sub>1/''n''</sub>}} does not have a zero measure. Thus there is some positive number {{mvar|c}} such that every countable collection of open intervals covering {{math|''X''<sub>1/''n''</sub>}} has a total length of at least {{mvar|c}}. In particular this is also true for every such finite collection of intervals. This remains true also for {{math|''X''<sub>1/''n''</sub>}} less a finite number of points (as a finite number of points can always be covered by a finite collection of intervals with arbitrarily small total length).
@@ Line 182: / Line 182: @@
 For every partition of {{math|[''a'', ''b'']}}, consider the set of intervals whose interiors include points from {{math|''X''<sub>1/''n''</sub>}}. These interiors consist of a finite open cover of {{math|''X''<sub>1/''n''</sub>}}, possibly up to a finite number of points (which may fall on interval edges). Thus these intervals have a total length of at least {{mvar|c}}. Since in these points {{mvar|f}} has oscillation of at least {{math|1/''n''}}, the infimum and supremum of {{mvar|f}} in each of these intervals differ by at least {{math|1/''n''}}. Thus the upper and lower sums of {{mvar|f}} differ by at least {{math|''c''/''n''}}. Since this is true for every partition, {{mvar|f}} is not Riemann integrable.
-We now prove the converse direction using the sets {{math|''X''<sub>''ε''</sub>}} defined above.<ref>[http://unapologetic.wordpress.com/2009/12/09/jordan-content-integrability-condition/ Jordan Content Integrability Condition], John Armstrong, December 9, 2009, The Unapologetic Mathematician</ref> For every {{mvar|ε}}, {{math|''X''<sub>''ε''</sub>}} is compact, as it is bounded (by {{mvar|a}} and {{mvar|b}}) and closed:
+We now prove the converse direction using the sets {{math|''X''<sub>''ε''</sub>}} defined above. For every {{mvar|ε}}, {{math|''X''<sub>''ε''</sub>}} is compact, as it is bounded (by {{mvar|a}} and {{mvar|b}}) and closed:
 *For every series of points in {{math|''X''<sub>''ε''</sub>}} that is converging in {{math|[''a'', ''b'']}}, its limit is in {{math|''X''<sub>''ε''</sub>}} as well. This is because every neighborhood of the limit point is also a neighborhood of some point in {{math|''X''<sub>''ε''</sub>}}, and thus {{mvar|f}} has an oscillation of at least {{mvar|ε}} on it. Hence the limit point is in {{math|''X''<sub>''ε''</sub>}}.
@@ Line 235: / Line 235: @@
 :<math>\lim_{a\to\infty} \int_{-a}^a f(x)\,dx.</math>
-For example, consider the sign function {{math|''f''(''x'') {{=}} sgn(''x'')}} which is 0 at {{math|''x'' {{=}} 0}}, 1 for {{math|''x'' > 0}}, and −1 for {{math|''x'' < 0}}. By symmetry,
+For example, consider the sign function {{math|''f''(''x'') = sgn(''x'')}} which is 0 at {{math|''x'' = 0}}, 1 for {{math|''x'' > 0}}, and −1 for {{math|''x'' < 0}}. By symmetry,
 :<math>\int_{-a}^a f(x)\,dx = 0</math>

@@ Line 1: / Line 1: @@
 == Licensing ==
 Content obtained and/or adapted from:
-* [http://mathonline.wikidot.com/riemann-stieltjes-integrals Riemann-Stieltjes Integrals from mathonline.wikidot.com] under a CC BY-SA license
+* [https://en.wikipedia.org/wiki/Riemann_integral Riemann Integral, Wikipedia] under a CC BY-SA license