U)such that X is contained in the union of Qi, the union of all Qk is contained in the union of X and Qk, and ∑W(Q)<6,∑(Q2)<M in which case the volume Ve(X) is(uniquely) defined as the limit of ∑V(Q2) as E-0 and Qk are required to have empty pair-wise intersections. A common alternative notation for Ve(X)is Ve(X p(a)dao x∈X When p= l, we get a definition of the usual (Lebesque) volume. It can be shown that the weighted volume is well defined for every compact subset of U, and also for every open subset of U for which the closure is contained in U. It is important to remember that not every bounded subset of U has a volume, even when p= 1 11.1.2 Volume change under a smooth map The rules for variable change in integration allow one to trace the change of weighted volume under a smooth transformation Theorem 11.1 Let u be an open subset ofr". Let F: U-u be an injective Lipschitz function which is differentiable on an open subset Uo of U such that the complement ofUo in U has zero Lebesque volume. LetP: UHR be a given measureable function which is bounded on every compact subset ofU. Then, if p-weighted volume is defined for a subset XCU, p-weighted volume is also defined for F(X, Pp-weighted volume is defined for X. where P(F(r)) det(dF/dr(i))l, dF/d r defined for i, and VP(F(XD)= VPF(X) Note that the formula is not always valid for non-injective functions(because of possi- ble "folding"). It is also useful to remember that image of a line segment(zero Lebesque volume when n> 1) under a continuous map could cover a cube (positive Lebesque volume)� � � � � 2 U) such that X is contained in the union of Qi k, the union of all Q2 is contained in the k union of X and Q1 k, and k) < �, V|�|(Q2 V|�|(Q1 k) < M, k k in which case the volume V�(X) is (uniquely) defined as the limit of V�(Q2 k) k as � � 0 and Q2 k are required to have empty pair-wise intersections. A common alternative notation for V�(X) is V�(X) = �(x)dx. x�X When � ≥ 1, we get a definition of the usual (Lebesque) volume. It can be shown that the weighted volume is well defined for every compact subset of U, and also for every open subset of U for which the closure is contained in U. It is important to remember that not every bounded subset of U has a volume, even when � ≥ 1. 11.1.2 Volume change under a smooth map The rules for variable change in integration allow one to trace the change of weighted volume under a smooth transformation. Theorem 11.1 Let U be an open subset of Rn. Let F : U ∞� U be an injective Lipschitz function which is differentiable on an open subset U0 of U such that the complement of U0 in U has zero Lebesque volume. Let � : U ∞� R be a given measureable function which is bounded on every compact subset of U. Then, if �-weighted volume is defined for a subset X � U, �-weighted volume is also defined for F(X), �F -weighted volume is defined for X, where �(F(x))| det(dF/dx(¯x))|, dF/dx defined for x, ¯ �F (¯x) = 0, otherwise, and V�(F(X)) = V�F (X). Note that the formula is not always valid for non-injective functions (because of possi￾ble “folding”). It is also useful to remember that image of a line segment (zero Lebesque volume when n > 1) under a continuous map could cover a cube (positive Lebesque volume)