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Riemann Integral, Measure Theor
and
Lebesgue Integral
By:
Tejan VadherNikunj Patel
Applie
Mathematics & Dept
NIT Sura
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Riemann Integral
Bernhard Riemann(1826 - 1866)
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Riemann Integral
Bernhard Riemann(1826 - 1866)
=
1
. (
) 1 <
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Riemann Integral
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Riemann Integral
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Riemann Integral
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Riemann Integral
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Riemann Integral
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Riemann Integral
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Riemann Integral
.1+.+
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Riemann Integral
(
0) +
1
+
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Riemann Integral
(
0) +
1
+
=
i=
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Shortcomings of Riemann Integration
1) .
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Shortcomings of Riemann Integration
1) .2)
.
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Shortcomings of Riemann Integration
1) .2)
.3) .
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Measure Theory
: , .
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Measure Theory
: , .
, 2, 3, , 9 1, 10
, 9., : 9.
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Measure Theory
?
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Measure Theory
? ,
, /{2, 3, , 9}
?
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Measure Theory
? ,
, /{2, 3, , 9}
9
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Measure Theory
? ,
, /{2, 3, , 9}
9 !
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Measure Theory
. .
.
.
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Measure Theory
:
, , .
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Measure Theory
:
, , . :
; , (/). .
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Measure Theory
:1) *
.
2) :
mA mA
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Measure Theory
:3) 1, 2, . . . }
,
=
=
()
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Lebesgue Integral
Henri Lebesgue(18751941)
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Lebesgue Integral
Henri Lebesgue(18751941)
0, 1, :
=
i
( | i1
() i1 i
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Lebesgue Integral
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Comparison of Riemann and Lebesgue Integral
.
.
()
=
[ ,]
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References:
www.ann.jussieu.fr/frey/cours/UdC/ma691/ma691_ann2.pdf
www.youtube.com/watch?v=ViqrHGae7FA
en.wikipedia.org/wiki/Outer_measure
en.wikipedia.org/wiki/Lebesgue_integration
collegemathteaching.wordpress.com/2011/01/17/integration-rieintegration-limitations-and-lebesgues-idea/
www.physicsforums.com/threads/lebesgue-vs-riemann-integral.
mathoverflow.net/questions/52708/why-should-one-still-teachintegration
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Thank you!
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