Lebesgue measure on the real line and on
We construct Lebesgue measure on the real line, as a particular case of Stieltjes measure associated
to the function
x ↦ x. We obtain as a consequence Lebesgue measure on
ℝⁿ. We prove that they
are translation invariant.
We show that, on
ℝⁿ, a linear map acts on Lebesgue measure by rescaling it through the absolute
value of its determinant, in
More properties of the Lebesgue measure are deduced from this in
haar_lebesgue.lean, where they
are proved more generally for any additive Haar measure on a finite-dimensional real vector space.
Definition of the Lebesgue measure and lengths of intervals #
Volume of a box in
Images of the Lebesgue measure under translation/multiplication in ℝ #
Images of the Lebesgue measure under translation/linear maps in ℝⁿ #
A diagonal matrix rescales Lebesgue according to its determinant. This is a special case of
real.map_matrix_volume_pi_eq_smul_volume_pi, that one should use instead (and whose proof
uses this particular case).
A transvection preserves Lebesgue measure.
Any invertible matrix rescales Lebesgue measure through the absolute value of its determinant.
Any invertible linear map rescales Lebesgue measure through the absolute value of its determinant.
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.