Properties of the Integral
Riemann integral and Lebesgue integral
Riemann integrability in its narrow meaning implies Lebesgue integrability.
Riemann integrability in its extended meaning does not imply Lebesgue integrability.
First mean value theorem of integration
Consider set
, and assume the following:
bounded,
integrable,
then
is integrable in 
.
satisfying
exists.
Lebesgue convergence theorem
Let
be 
-measurable in 
.If a function
that is integrable in 
satisfies 
at every point of 
, then
If
, then
Integral and derivative
Consider a function
in measure space 
that is integrable in 
and differentiable in 
.If a function
is integrable in 
and satisfies 
for 
, then
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