|mA| = |detm| |A| For every Lebesgue set A belogning to R^n and every invertible n x n matrix m. Use this to prove the change of variable theorem.
(*) The integral (mb) of f(y)dy = the integral (b) of f(mx)|detm| dx for everyy Lebesgue measurable function f = R^n --> R which is LEbesgue integrable on a Lebesgue set B the union of R^n.
Hint: First establish (*) for simple functions f = SUM (C_k)(X_Ak)
with |A_k| < infinity for all k. Recall that the integral (mb) of f(y) dy means the integral (R^n) of fX_mb du_m where u_n is Lebesgue measure on R^n.© BrainMass Inc. brainmass.com March 4, 2021, 5:55 pm ad1c9bdddf
A Lebesgue set is investigated and the Change of Variable theorem is proven. Lebesgue integrability is also investigated The solution is provided in an attached Word document.