Lebesgue Set: Change of Variable Theorem and Integrability
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|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.
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Solution Summary
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.
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