测度论引论 (英文版)

原文摘录   · · · · · · 

• "We note that once one adopts the convention $+\infty \cdot 0 =0 \cdot +\infty =0 $,then multiplication becomes upward continuous (in the sense that whenever $x_n \in [0,+\infty]$ increases to $x\in [0,+\infty]$,and $y_n \in [0,+\infty]$ increases to $y\in [0,+\infty]$,then $x_n y_n$ increases to $xy$) but not downward continuous(e.g. $1/n\rightarrow 0$ ,$1/n \cdot \infty\rightarrow \infty$). 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 1.4.44) applies for monotone increasing functions,but not necessarily for monotone decreasing ones.)" (查看原文)

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