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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!