可測函數 — JOVANA Education

定義：射線的原像

正如連續性由開集的原像刻畫，可測性由射線的原像刻畫。函數 f : ℝ → ℝ 稱為 [[measurable-function|可測的]]，若對每個實數 a，集合 {x : f(x) > a} 都可測。我們只需用到射線 (a, ∞)；其餘方向的射線，以及介於其間的一切區間與波萊爾集，隨即由補、可數並與交自動隨之成立。

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.

由一種射線推及全部：一條假設即可控制每個下水平集與上水平集。

封閉性：極限再也無法擊破的性質

這正是與第一篇形成決定性對比之處。可測函數對一切運算封閉：若 f、g 可測，則 f + g、fg、|f|、max(f,g)、cf 皆可測。關鍵在於，若 f_1, f_2, … 全可測，則 sup f_n、inf f_n、lim sup f_n、lim inf f_n 以及任意逐點極限 lim f_n 也都可測。取極限時這一類不會「漏出去」——而那正是斷送黎曼積分的那種精確失敗。

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.

上確界、下確界、上極限、下極限與逐點極限都留在可測類之內。

構件：簡單函數

最簡單的可測函數是簡單函數：有限組合 c_1·1_{A_1} + … + c_n·1_{A_n}，其中每個示性函數 1_{A} 在可測集 A 上取 1、在其外取 0。簡單函數只取有限多個值，且每個水平集可測。它們是分析學家的「階梯函數」，但建立在任意可測集而非區間之上。

作為基石的逼近定理表明它們已經足夠：每個非負可測函數 f 都是簡單函數的遞增逐點極限 0 ≤ s_1 ≤ s_2 ≤ … ↗ f。構造 s_n 的方法是把值域 [0, n] 切成 2^n·n 條細的水平帶，再把 f 向下取整到它所落入那條帶的底端。這恰是第一篇中「切分值域」的想法，如今得到精確化——它正是通往勒貝格積分的門戶：下一階梯將先對簡單函數積分、再取極限，從而構造該積分。