14.5. MEASURABLE FUNCTIONS 333

Proof: Let f+ (ω) ≡ | f (ω)|+ f (ω)2 , f− (ω) ≡ | f (ω)|− f (ω)

2 . Thus f = f+− f− and | f | =f++ f−. Also f = f+ when f ≥ 0 and f =− f− when f ≤ 0. Both f+, f− are measurablefunctions. Indeed, if a≥ 0, f−1

+ ((a,∞)) = f−1 ((a,∞)) ∈F . If a < 0 then f−1+ ((a,∞)) =

Ω. Similar considerations hold for f−. Now let s+n (ω) ↑ f+ (ω) ,s−n (ω) ↑ f− (ω) meaningthese are simple functions converging respectively to f+ and f− which are both increasingin n and nonnegative. Thus if sn (ω)≡ s+n (ω)−s−n (ω) , this converges to f+ (ω)− f− (ω) .Also

|sn (ω)|= s+n (ω)+ s−n (ω)≤ f+ (ω)+ f− (ω) = | f (ω)|

Definition 14.5.9 Rp consists of the mappings from (1,2, · · · , p) to R. We usuallywrite it as follows. x ∈ Rp means x is an ordered list of p real numbers. Thus

x = (x1, · · · ,xp)

(1,2,3) is in R3. Note that (1,2,3) ̸= (3,2,1) because the two have different real num-bers in some locations. In terms of functions, x(1) ̸= x(3).

Definition 14.5.10 If xn =(xn

1, · · · ,xnp)

is a sequence of points in Rp, then we saylimn→∞ xn = x if and only if limn→∞ xn

i = xi for each i. In other words, convergence takesplace if and only if the component entries of xn converge to the corresponding componententries of x. We say that g : D→R is continuous for D⊆Rp if whenever xn→ x with eachxn ∈ D and x ∈ D, then g(xn)→ g(x) .

In other words, it is essentially the same as what was presented earlier for continuousfunctions of one variable.

Proposition 14.5.11 Let fi : Ω→ R be measurable, (Ω,F ) a measurable space, andlet g : Rp→ R be continuous. If f(ω) =

(f1 (ω) · · · fp (ω)

)T, then g◦ f is measur-

able.

Proof: From Corollary 14.5.8 above, there are sni (ω) , simple functions

limn→∞

sni (ω) = fi (ω)

such that |sni (ω)| ≤ | fi (ω)|. Let sn (ω) ≡

(sn

1 (ω) · · · snp (ω)

)thus, by continuity,

g(sn (ω))→ g(f(ω)) for each ω. It remains to verify that g◦ sn is measurable.

(g◦ sn)−1 (a,∞)≡ {ω : g(sn (ω))> a} .

This is the finite union of measurable sets since each sni is a simple function having finitely

many values. Thus there are finitely many possible values for g ◦ sn, each value corre-sponding to the intersection of p measurable sets. Therefore, g ◦ sn is measurable. ByCorollary 14.5.4, it follows that g◦ f , being the pointwise limit of measurable functions isalso measurable.

Note how this shows as a very special case that linear combinations of measurable realvalued functions are measurable because you could take g(x,y) ≡ ax+by and then if youhave two measurable functions f1, f2, it follows that a f1 +b f2, f1 f2 are measurable. Also,if f is measurable, then so is | f |. Just let g(x) = |x|. In general, you can do pretty muchany algebraic combination of measurable functions and get one which is measurable. Thisis very different than the case of generalized integrable functions.