My class notes on Stochastic Calculus says that processes in $\mathbb{M_c}^{loc}, \mathbb{A}_c $ and $\mathbb{V}_c$ where they have their usual meaning, are indistinguishable of continuous processes. What does this mean? I know that if a two processes are right/left continuous and they are modifications of each other then they are indistinguishable. But I do not understand how can processes in say $\mathbb{M_c}^{loc}$ be indistinguishable of any random continuous processes. I am probably missing something here. Could someone explain it to me?
Edit: I am adding the definitions of $\mathbb{M_c}^{loc}, \mathbb{A}_c $ and $\mathbb{V}_c$
Where $\mathbb{M_c}^{loc}=\{\text{All continuous local martingales where } M_0=0 \}$
$\mathbb{A}_c=\{A\mid \text{A is adapted and a.s continuous and increasing process with } A_0=0\}$
$\mathbb{V}_c=\{A\mid \text{A is adapted and a.s continuous and finite variation process with } V_0=0\}$