Let $B$ be a Brownian motion. Definining a *pathwise* stochastic integral $I(f):=\int f~dB$ for certain classes of deterministic functions is straightforward: For instance if $f=\sum_ic_i1\{[t_i,t_{i+1})\}$ is a compactly supported simple function, then
$$I(f):=\sum_ic_i(B(t_{i+1})-B(t_i)).$$
More generally if $AC_0$ denotes the set of compactly supported absolutely continuous functions, continuity of $B$ allows to define the stochastic integral via integration by parts
$$I(f):=-\int f'(x)B(x)~dx.\tag{1}$$

Two properties of this "stochastic integral operator" on $AC_0$ that are of interest to me are:

 1. The operator defines a *pathwise* stochastic integral, i.e., for almost every outcome $\omega\in\Omega$ in our probability space, the brownian path $B_\omega\in C[0,\infty)$ yields a linear functional $I_\omega:AC_0\to\mathbb R$, as $(1)$ satisfies
$$I_\omega(af+bg)=aI_\omega(f)+bI_\omega(g).$$
 3. The random functional $f\mapsto I(f)$ is a centered Gaussian indexed by $AC_0$ process with covariance given by the $L^2$ inner product.

> **Question.** Is there a way to extend this operator to deterministic functions with much less regularity (namely, continuous with compact support) *while preserving the above two properties simultaneously*?

Using Itô's theory, we can of course define $I(f)\in L^2(\Omega)$ for every $f\in L^2[0,\infty)$, but my understanding is that this construction fails the above question. That is, for any fixed $f,g\in L^2$, we have linearity in the sense that
$$I(af+bg)=aI(f)+bI(g)\tag{2}$$
as elements of $L^2(\Omega)$, but this is a *weaker* linearity: the probability-one event on which $(2)$ holds depends on $a,b,f,g$, and thus from this alone we can't define $I$ as an almost sure linear operator $I_\omega:L^2[0,\infty)\to\mathbb R$.