24.1. STRONG AND WEAK MEASURABILITY 651

What of limits of measurable functions? The next theorem says that the usual theo-rem about limits of measurable functions being measurable holds. The proof is similar toshowing that the limit of measurable finitely valued functions is measurable given above.

Theorem 24.1.10 Let xn and x be functions mapping Ω to X where F is a σ al-gebra of measurable sets of Ω and X is a Banach space. Thus X satisfies 24.1. Thenif xn is strongly measurable, and x(ω) = limn→∞ xn(ω), it follows that x is also stronglymeasurable. (Pointwise limits of measurable functions are measurable.)

Proof: Let {Vm} be the sequence of 24.1. Since x is the pointwise limit of xn,

x−1(Vm)⊆ {ω : xk(ω) ∈Vm for all k large enough} ⊆ x−1(Vm).

Therefore,x−1(V ) = ∪∞

m=1x−1(Vm)⊆ ∪∞m=1∪∞

n=1∩∞k=nx−1

k (Vm)

⊆ ∪∞m=1x−1(Vm) = x−1(V ).

It follows x−1(V ) ∈F because it equals the expression in the middle which is measurable.Note that this shows the characterization of measurability in terms of inverse images ofopen sets being measureable sets. Thus the theorem is proved in the case of separableBanach spaces. However, Lemma 24.1.3 can be applied to conclude that this holds ingeneral because each xn is separably valued given they are each strongly measurable andx(Ω)⊆ D where D = ∪nDn for Dn a countable dense subset of xn (Ω). ■

Note that the same conclusion in terms of inverse images being measurable would holdfor any metric space.

Corollary 24.1.11 x is strongly measurable if and only if x(Ω) is separable and x isweakly measurable.

Proof: Strong measurability clearly implies weak measurability. If xn (ω) → x(ω)where xn is simple, then f (xn (ω))→ f (x(ω)) for all f ∈ X ′. Hence f ◦ x is measurableby Theorem 24.1.10 because it is the limit of a sequence of measurable functions. Let Ddenote the set of all values of the xn. Then D is a separable set containing x(Ω). Thus D isa separable metric space. Therefore x(Ω) is separable also by the last part of the proof ofTheorem 24.1.5.

Now suppose D is a countable dense subset of x(Ω) and x is weakly measurable. LetZ be the subset consisting of all finite linear combinations of D with the scalars comingfrom the set of rational points of F. Thus, Z is countable. Letting Y = Z, Y is a separableBanach space containing x(Ω). If f ∈ Y ′, f can be extended to an element of X ′ by theHahn Banach theorem. Therefore, x is a weakly measurable Y valued function. Now useTheorem 24.1.8 to conclude x is strongly measurable. ■

Weakly measurable as defined above means ω → x∗ (x(ω)) is measurable for everyx∗ ∈ X ′. The next lemma ties this weak measurability to the usual version of measurabilityin which a function is measurable when inverse images of open sets are measurable.

Lemma 24.1.12 Let X be a Banach space and let x : (Ω,F )→ K ⊆ X where K isweakly compact and X ′ is separable. Then x is weakly measurable if and only if x−1 (U) ∈F whenever U is a weakly open set.