Notation xi The extended non-negative real axis [0, +∞] is the non-negative real axis [0, +∞) := {x R : x 0} with an additional element adjointed to it, which we label +∞ we will need to work with this system because many sets (e.g. Rd) will have infinite measure. Of course, +∞ is not a real number, but we think of it as an extended real number. We extend the addition, multiplication, and order structures on [0, +∞) to [0, +∞] by declaring +∞ + x = x + +∞ = +∞ for all x [0, +∞], +∞ · x = x · +∞ = +∞ for all non-zero x (0, +∞], +∞ · 0 = 0 · +∞ = 0, and x +∞ for all x [0, +∞). Most of the laws of algebra for addition, multiplication, and order continue to hold in this extended number system for instance, addition and multi- plication are commutative and associative, with the latter distributing over the former, and an order relation x y is preserved under addition or mul- tiplication of both sides of that relation by the same quantity. However, we caution that the laws of cancellation do not apply once some of the vari- ables are allowed to be infinite for instance, we cannot deduce x = y from +∞ + x = +∞ + y or from +∞· x = +∞· y. This is related to the fact that the forms +∞ +∞ and +∞/ + are indeterminate (one cannot assign a value to them without breaking many of the rules of algebra). A general rule of thumb is that if one wishes to use cancellation (or proxies for cancel- lation, such as subtraction or division), this is only safe if one can guarantee that all quantities involved are finite (and in the case of multiplicative can- cellation, the quantity being cancelled also needs to be non-zero, of course). However, as long as one avoids using cancellation and works exclusively with non-negative quantities, there is little danger in working in the extended real number system. We note also that once one adopts the convention +∞ · 0 = 0 · +∞ = 0, then multiplication becomes upward continuous (in the sense that when- ever xn [0, +∞] increases to x [0, +∞], and yn [0, +∞] increases to y [0, +∞], then xnyn increases to xy) but not downward continuous (e.g. 1/n 0 but 1/n · +∞ 0 · +∞). This asymmetry will ultimately cause us to define integration from below rather than from above, which leads to other asymmetries (e.g. the monotone convergence theorem (Theorem 1.4.43) applies for monotone increasing functions, but not necessarily for monotone decreasing ones).
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