Lebesgue measure on the real line and on
Preliminary definitions #
Definition of the Lebesgue measure and lengths of intervals #
Lebesgue measure on the Borel sets
The outer Lebesgue measure is the completion of this measure. (TODO: proof this)
Volume of a box in
Images of the Lebesgue measure under translation/multiplication/... #
The volume of the region between two almost everywhere measurable functions on a measurable set can be represented as a Lebesgue integral.
If two functions are integrable on a measurable set, and one function is less than or equal to the other on that set, then the volume of the region between the two functions can be represented as an integral.