Measurable Functions

The definition: preimages of rays

Just as continuity is captured by preimages of open sets, measurability is captured by preimages of rays. A function f : ℝ → ℝ is [[measurable-function|measurable]] if for every real number a the set {x : f(x) > a} is measurable. We only need the rays (a, ∞); rays in the other directions, and all the intervals and Borel sets in between, then follow automatically by complement, countable union, and intersection.

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.

From one type of ray to all of them: a single hypothesis controls every sublevel and superlevel set.

Closure: the property limits could not break before

Here is the decisive contrast with Guide 1. The measurable functions are closed under everything: if f and g are measurable, so are f + g, fg, |f|, max(f,g), and cf. Crucially, if f_1, f_2, … are all measurable, then sup f_n, inf f_n, lim sup f_n, lim inf f_n, and any pointwise limit lim f_n are measurable too. The class does not leak when you take limits — the precise failure that doomed the Riemann integral.

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.

Sup, inf, limsup, liminf, and pointwise limits all stay inside the measurable class.

Building blocks: simple functions

The simplest measurable functions are the simple functions: finite combinations c_1·1_{A_1} + … + c_n·1_{A_n}, where each characteristic function 1_{A} is 1 on a measurable set A and 0 off it. A simple function takes only finitely many values, and each level set is measurable. These are the analysts' “step functions,” but built on arbitrary measurable sets rather than intervals.

The cornerstone approximation theorem says these are enough: every non-negative measurable function f is the increasing pointwise limit of simple functions 0 ≤ s_1 ≤ s_2 ≤ … ↗ f. You build s_n by chopping the range [0, n] into 2^n·n thin horizontal strips and rounding f down to the bottom of whichever strip it lands in. This is exactly the range-slicing idea from Guide 1, now made precise — and it is the doorway to the Lebesgue integral, which the next track will build by integrating simple functions and passing to the limit.