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This question comes from proof of Theorem 2.47 in Folland's "real analysis: modern techniques and their applications", second edition. In particular, the question lies in the inequalities in line 7 and 8 in page 76. The first equality is an application of measure property "continuity from above". But for this property to be applicable, m(G(U_1)) (or m(G(U_j)) for at least one of j>1) must be finite. If we can prove that \int_{U_1}|\det D_x G|dx is finite, we can derive this result by the inequality in line 3. Secondly, to use dominated convergence theorem to obtain the last "=", |\det D_x G| must be integrable on U_j, that is, \int_{U_j}|\det D_x G|dx must be finite. But I can not prove this. I tried to show that |\det D_x G| is bounded on U_j, but U_j is only an open set, although it has finite measure. Could you please help me prove that the integral \int_{U_j}|\det D_x G|dx is finite? Thanks!
The following image contains part of this proof, for reference.
The following image contains part of this proof, for reference.