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## Main Question or Discussion Point

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, [tex]m(G(U_1))[/tex] (or [tex]m(G(U_j))[/tex] for at least one of

The following image contains part of this proof, for reference.

*j*>1) must be finite. If we can prove that [tex]\int_{U_1}|\det D_x G|dx[/tex] is finite, we can derive this result by the inequality in line 3. Secondly, to use dominated convergence theorem to obtain the last "=", [tex]|\det D_x G|[/tex] must be integrable on [tex]U_j[/tex], that is, [tex]\int_{U_j}|\det D_x G|dx[/tex] must be finite. But I can not prove this. I tried to show that [tex]|\det D_x G|[/tex] is bounded on [tex]U_j[/tex], but [tex]U_j[/tex] is only an open set, although it has finite measure. Could you please help me prove that the integral [tex]\int_{U_j}|\det D_x G|dx[/tex] is finite? Thanks!The following image contains part of this proof, for reference.