可测函数 — JOVANA Education

定义：射线的原像

正如连续性由开集的原像刻画，可测性由射线的原像刻画。函数 f : ℝ → ℝ 称为 [[measurable-function|可测的]]，若对每个实数 a，集合 {x : f(x) > a} 都可测。我们只需用到射线 (a, ∞)；其余方向的射线，以及介于其间的一切区间与波莱尔集，随即由补、可数并与交自动随之成立。

Claim: if {f > a} is measurable for every a, then {f >= a} is too.

   {x : f(x) >= a} = INTERSECTION over n=1,2,3,...  of  {x : f(x) > a - 1/n}.

  Why: f(x) >= a  iff  f(x) > a - 1/n for every n
       (the strict-inequality sets shrink down to the closed condition).
  Each {f > a - 1/n} is measurable by hypothesis;
  a COUNTABLE intersection of measurable sets is measurable (sigma-algebra).
  Hence {f >= a} is measurable.

Similarly:
   {f < a} = complement of {f >= a}              -> measurable
   {f <= a} = complement of {f > a}               -> measurable
   {a < f < b} = {f > a} intersect {f < b}        -> measurable
   {f = a} = {f <= a} intersect {f >= a}          -> measurable
So ANY of these four ray-conditions could serve as the definition; all agree.

由一种射线推及全部：一条假设即可控制每个下水平集与上水平集。

封闭性：极限再也无法击破的性质

这正是与第一篇形成决定性对比之处。可测函数对一切运算封闭：若 f、g 可测，则 f + g、fg、|f|、max(f,g)、cf 皆可测。关键在于，若 f_1, f_2, … 全可测，则 sup f_n、inf f_n、lim sup f_n、lim inf f_n 以及任意逐点极限 lim f_n 也都可测。取极限时这一类不会“漏出去”——而那正是断送黎曼积分的那种精确失败。

Why the supremum of measurable functions is measurable.
  Let g(x) = sup_n f_n(x), each f_n measurable.
  Key identity:
      {x : g(x) > a} = UNION over n  of  {x : f_n(x) > a}.
  Why: g(x) > a  iff  the sup exceeds a  iff  SOME f_n(x) > a.
  Right side is a countable union of measurable sets -> measurable.
  So g = sup f_n is measurable.

Then, building up:
  inf_n f_n = - sup_n (-f_n)                         -> measurable
  limsup_n f_n = inf_k ( sup_{n>=k} f_n )            -> measurable
  liminf_n f_n = sup_k ( inf_{n>=k} f_n )            -> measurable
  If lim f_n exists pointwise, lim f_n = limsup f_n  -> measurable

Contrast Guide 1: there, pointwise limits ESCAPED Riemann integrability.
Here the class is sealed shut under exactly that operation.

上确界、下确界、上极限、下极限与逐点极限都留在可测类之内。

构件：简单函数

最简单的可测函数是简单函数：有限组合 c_1·1_{A_1} + … + c_n·1_{A_n}，其中每个示性函数 1_{A} 在可测集 A 上取 1、在其外取 0。简单函数只取有限多个值，且每个水平集可测。它们是分析学家的“阶梯函数”，但建立在任意可测集而非区间之上。

作为基石的逼近定理表明它们已经足够：每个非负可测函数 f 都是简单函数的递增逐点极限 0 ≤ s_1 ≤ s_2 ≤ … ↗ f。构造 s_n 的方法是把值域 [0, n] 切成 2^n·n 条细的水平带，再把 f 向下取整到它所落入那条带的底端。这恰是第一篇中“切分值域”的想法，如今得到精确化——它正是通往勒贝格积分的门户：下一阶梯将先对简单函数积分、再取极限，从而构造该积分。