# Some basics for vector-valued L^p spaces, when p is finite

There is a well-established way to extend the familiar definition of the Lebesgue spaces (for 1 ≤ *p* ≤ ∞) to the setting of *E*-valued functions on , where *E* is a complex-valued Banach space. I say “well-established” but it was only relatively recently (c. 2013/2014) that I found myself needing to look up some of the technical subtleties, and even more recently I realised I had remained ignorant of some fairly important foundational aspects of this process or construction.

When 1≤ *p* < ∞ the natural guess is that we should take equivalence classes of those “measurable” functions that satisfy

But we run into an immediate issue: which functions do we declare to be "measurable"? It is clearly necessary for to be measurable as a function (as well as *p*-integrable), but is this a good definition or should it be treated as a consequence of some “better” definition?

Moreover: how should we treat the case *p*=∞ ?

## Digression on Borel versus Lebesgue

We shall work with the Borel sigma-algebra on [0,1] rather than the Lebesgue sigma-algebra , though this is largely a matter of preference/convenience: see this EOM entry for some of the relevant context. In the sense of these lecture notes by J. van Neerven, we are working with (strongly or weakly) -measurable functions, rather than (strongly or weakly) μ-measurable functions where μ is taken to be Lebesgue measure.

## The key definitions

**Definition 1.** A function is called a *simple function* if there are (Borel) measurable subsets and in such that

**Definition 2.** A function is said to be *strongly measurable* if there is a sequence of simple E-valued functions such that for all .

**Definition 3.** A function is said to be *weakly measurable* if, for each , the scalar-valued function is (Borel-to-Borel) measurable.

**Definition 4.** A function is said to be *Borel measurable* if is Borel for every Borel subset .

It is a straightforward exercise to show that if is a sequence of Borel measurable functions and pointwise, then is Borel measurable. See e.g. this MO answer. The same MO answer also explains why strongly measurable *E*-valued functions are both weakly measurable and Borel measurable. (Essentially, one verifies this for simple functions, and then invokes the result mentioned above.)

The image of any strongly measurable function is always separable (with respect to the norm topology of *E*), and so if is strongly measurable, it is weakly measurable and takes values in a separable subspace of *E*. An important theorem of Pettis tells us that the converse is true.

**Theorem 5 (Pettis measurability theorem).** Let be a weakly measurable function which takes values in a separable subspace of *E*. Then is strongly measurable.

(One can weaken the hypotheses slightly: it is enough to require separable range and "σ(*E*,*F*)-measurability", where *F* is a weak-star dense subspace of *E*^{*}. See the notes of van Neerven that are linked above.)

Note that if is any function between two Banach spaces, *not necessarily linear*, and is a simple function, then is also a simple function. With this in mind, the following result is an almost immediate consequence of Definition 1.

**Proposition 6.** Let *E* and *F* be Banach spaces and let be a *continuous* function. If is strongly measurable, then so is .

In particular, if is strongly measurable, then is (Borel-to-Borel) measurable.

**Remark 7.** Proposition 6 may seem unsurprising, given that the corresponding statement where “strongly” is replaced by “Borel” is extremely easy to prove. However, it’s not clear to me right now if the result still holds when “strongly” is replaced by “weakly”.

## Lebesgue-Bochner spaces for finite *p*

In view of the comment following Proposition 6, we can now define *E*-valued in a natural way.

**Definition 8.** Let *E* be a Banach space. We say that a strongly measurable function is *Bochner integrable* if . The vector space *V* of all such carries an obvious/natural seminorm, and we define to be the quotient of *V* by the kernel of this seminorm.

We view elements of to be equivalence classes of Bochner integrable *E*-valued functions, with the equivalence relation being “differs only on a set of measure zero”.

Loosely speaking, for 1≤*p*< ∞, we can then define to be the space of all (equivalence classes of) strongly measurable functions for which is integrable. Equipping with the natural norm, one can show it is complete by adapting the usual proofs from the scalar-valued case.

It is not hard to prove that the simple functions form a norm-dense subspace of , and so if one likes tensor product formalism one can view as a completion of the algebraic tensor product .

## But what about *p*=∞?

This will be handled in ~~the next~~ some future blog post.