$f: [0,1]\rightarrow L^1(\Omega)$ as a (measurable?) function from $[0,1]\times \Omega\rightarrow \mathbb{R}$

Question

Given a map from $\big([0,1], \mathcal{B}[0,1], m\big)$ to a Banach space $(X, \|\cdot \|)$. There are strong measurable functions (they are the point wise a.e. limit of simple functions) and weak measurable functions (for each $u^* \in X^*$, we have $t\mapsto \langle u^*, f(t)\rangle$ is a measurable as a function from $[0,1] \rightarrow \mathbb{R}$).

I have two questions:

1. Why don't we use the normal measurable definition here? That is a function is measurable if the pre-image of all Borel sets $U\in \mathcal{B}(X)$ are in $\mathcal{B}[0,1]$.
2. Now suppose $X= L^1(\Omega, \mathcal{F}, \mu)$, given a measurable $f:[0,1]\rightarrow L^1$ (either strong, weak or the normal definition), can we say that the function $f(t,\omega)$ is measurable as a function from $[0,1] \times \Omega \rightarrow \mathbb{R}$ with respect to the product sigma algebra on the domain?

Here, assume elements of $L^1$ are just measurable functions, not equivalent classes, so that $f(t,x)$ is well defined.

Thank you for your time.

Answer 1

(1) If the Banach space $X$ is separable; and if you use the Lebesgue-measurable sets on $[0,1]$ not the Borel sets; then all three definitions are equivalent.

But of course the main thing of interest is not "measurable function" but "integrable function". When $X$ is not separable, you probably want the Bochner integral, using the definition pointwise a.e. limit of simple functions. There is also the Pettis integral, using weak measurability, but its properties are much worse than the Bochner integral.

For (2), it is clear with the Bochner definition of measurable: pointwise a.e. limit of simple functions. Again you want a complete measure (like Lebesgue) so that your sequnce can do weird things on a set of measure zero.

Plug:

• Edgar, G. A. Measurability in a Banach space. Indiana Univ. Math. J. 26 (1977), no. 4, 663–677. MSN.

• Edgar, G. A. Measurability in a Banach space. II. Indiana Univ. Math. J. 28 (1979), no. 4, 559–579. MSN.