[分析] 關於Measurable function

[ericakk (代買IPSA單品8折)]

在看Bartle的實變 對於課文看不太懂,請看黃色部分:

1.If Χ=Ｒ and Χ=Ｂ, then any monotone function is Borel measurable.

  For ,suppose that f is monotone increasing in the sense that x≦x',

  implies f(x) ≦ f(x'). Then {x屬於Ｒ:f(x)＞α} consists of a half-line

  which is either of the form {x屬於Ｒ:x＞α} or the form {x屬於Ｒ:x≧α},

  or is Ｒ or 空集合.

黃色部分在說什麼? 可以用數學造詣翻譯給我聽嗎?

2.Observe that if f 屬於 M(χ,Χ),then

                         ∞
  {x屬於χ:f(x) = +∞} = ∩{x屬於χ:f(x)＞n}
                        n=1

                                            ∞
  {x屬於χ:f(x) = －∞} = the complment of [∪{x屬於χ:f(x)＞－n}]
                                           n=1

請問黃色的＞符號,在這裡可以換成≧嗎?

謝謝回答^^

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