Lebesgue Integral in SDE

Question

In the context of proving existence of solutions of S(P)DEs, I've found that few (if any) texts offer significant mention to the deterministic drift term of the form $$ \int_0^t f(s,X(s))ds. $$ If we are to prove existence by the Picard iteration, it will be necessary to show that this integral, as a piece of the fixed point map, exists and has the requisite properties, such as predictability.

I was hoping for a critique (or a suggested reference) for my take on the following simpler problem:

Let $X$ be a predictable $E$ valued process, with $E$ a separable Banach space and $$ \sup_{t\leq T}\mathbb{E}[\|X(t)\|]<\infty $$ I want to prove that $Y(t) = \int_0^t X(s)ds$ exists (at least a.s.) and is predictable.

What I've come up with so far is the following:

1. Since $X$ is predictable, it is progressively measurable. Thus, for any $t$, it is $\mathcal{B}([0,t])\times \mathcal{F}_t$ measurable. Since $E$ is separable, $X$ will be strongly measurable. Since $\sup_t\mathbb{E}[\|X(t)\|]<\infty$, we can use Fubini to get the existence of the Bochner integral over both the probability space and $[0,t]$. Then, by Fubini (again), we get that $Y(t)$ exists a.s. and is clearly $\mathcal{F}_t$ measurable (i.e., it is adapted).

2. To get predictability, my instinct is to demonstrate that $Y$ is continuous in some sense, and then use that continuous + adapted implies it has a predictable version. It seems that we would want to write $$ \mathbb{E}[\|Y(t+h) - Y(t)\|]\leq \int_{t}^{t+h}\mathbb{E}[\|X(s)\|]ds\leq h\times \sup_{t}\mathbb{E}[\|X(t)\|] $$ But because of the way we have defined $Y$ (via the Bochner version of Fubini), my sense is that we need to do a bit more to treat this as a classical integral.
   Here, my first alternative approach is to instead define $$ Y(t) = \int_0^T \chi_{[0,t]}(s) X(s)ds $$ and then interpret this $Y(t) = \int_0^t X(s)ds$. This would then mean that $Y(t+h)-Y(t)$ would be an integral over a common set, $[0,T]$. Is this right?
   Is there another, simpler, argument?

3. It is then a standard argument that continuity in this sense + adaptedness would imply a predictable version exists. In the context of proving existence of the solutions by Picard iteration, is it satisfactory to interpret the fixed point map as giving the predictable version each time it is applied?