Let $(B_t)_{t\ge 0}$ be a real-valued Brownian motion on a probability space $(\Omega,\mathcal A,\operatorname P)$ and $\lambda$ be the Lebesgue measure on $\mathbb R$. Is $$W(\phi):=\int_{[0,\infty)}\phi(t)B_t\;{\rm d}\lambda\;\;\;\text{for }\phi\in C_c^\infty(\mathbb R)$$ well-defined?
Let $\phi\in C_c^\infty(\mathbb R)$ and $K:=\operatorname{supp}\phi\cap [0,\infty)$. I would argue that $$\left\|\phi1_{[0,\infty)}B(\omega,\;\cdot\;)\right\|_{L^1(\lambda)}\le\underbrace{\left\|\phi\right\|_{L^\infty(\lambda)}}_{<\infty}\underbrace{\left\|1_KB(\omega,\;\cdot\;)\right\|_{L^1(\lambda)}}_{=:M(\omega)}\;\;\;\text{for all }\omega\in\Omega$$ and that $M<\infty$ almost surely, since $K$ is compact and $B$ is almost surely continuous.
How do I deal with the null set on which $B$ is not continuous? Please note that $W$ is the distributional stochastic process associated with $B$. Maybe we don't care about its values on a null set.
$M<\infty$ because $B$ is almost surely continuous and thus bounded when restricted to a compact set