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Riemann integral (partitioning of the function domain): $$ \int_a^b fdx = \lim_{n\rightarrow\infty} \sum_{k=1}^n \inf_{[x_{i-1}, x_i]}f(x)d\mu([x_{i-1}, x_i]) $$

Lebesgue integral (partitioning of the function range): $$ \int_E fd\mu = \lim_{n\rightarrow\infty} \sum_{k=1}^n \inf_{E_k} f(x) \mu(E_k) $$

Questions:

1) Are the definitions above correct?

2) How to derive these definitions and what is the logic behind them?

Thank you for your help in advance!

Konstantin
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    An answer to a similar question might be helpful for 2): http://math.stackexchange.com/questions/1642034/eli5-riemann-integrable-vs-lebesgue-integrable/1642090#1642090 – Roland Mar 04 '17 at 11:59
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    Are the definitions correct? Sort of. If $f$ is integrable then those equations are true. But the limit on the right may exist even if $f$ is not integrable. – GEdgar Mar 04 '17 at 15:03
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    What you have written is not "partition the domain" vs. "partition the range" ... You have written two different schemes to partition the domain: into intervals vs. into measurable sets. See the answer linked by Roland for explanation of the "partition the range" scheme for the Lebesgue integral. – GEdgar Mar 04 '17 at 15:06

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