A question in a proof on approximating n-dimensional Lebesgue measurable set by open set

Question

In Folland's "real_analysis: modern techniques and their applications", second edition, page 70, line 4 of proof of Theorem 2.40 (a), the author asserts the inequality $m(U_j) < m(T_j)+\epsilon2^{-j}$. I think this comes from $m(i$-th side of $U_j)\leq m(i$-th side of $T_j)+$a small number, $i=1,...,n$, by Theorem 1.18. But I find if one of the sides of $T_j$ (assume $T_{j_1}$) has measure 0 and another side (assume $T_{j_2}$) has measure $+\infty$, $m(T_j)$ would still be 0 by the convention $0\cdot\infty=0$, but on the other hand the measure of the side $U_{j_1}$ coresponding to the null side may be nonzero since the elements can be all nonzero even if the infimum is 0, and the side $U_{j_2}$ corresponding to the infinite side satisfies $m(U_{j_2})=\infty$, thus $m(U_j)$ would be $\infty$, then the inequality can never hold. I can not amend this problem by, say, simply discarding this special kind of rectangles. This conclusion is right, but so far as I am concerned, it was established by a rather complicated process, as in Bartle's "The Elements of Integration and Lebesgue Measure"([B]) for example, first build up from cells ([B]P132,Definition 12.1), then reduce from general cells to open cells([B]P133, Remarks 12.2(b)), then prove that this construction is identical to the product measure approach as Folland ([B]P149 Lemma 14.1 and uniqueness of extension) and finally prove it using the similar method([B]P155-156 Lemma 15.1(a)). So my question is, 1)if it is an error? 2)if it is, how to amend it? do we have to use the complicated process mentioned above? Thanks! The following image is a snapshot of the proof, there are a few typoes in it: the "$R_j$" and "$F_j$" in line 3 and 4, respectly, are actually "$T_j$". alt text http://i52.tinypic.com/33e1mxd.png

Answer 1

If some rectangle of measure zero has a side of infinite measure, divide that side into a countable number of sides of finite measure. In other words, you may arrange things so that none of the $T_j$ has any side of infinite measure.