I'm reading about continuous semi-martingale from page 8 of these notes.
Let $(\Omega, \mathcal F, \mathbb P)$ be a probability space.
Definition. A continuous semi-martingale w.r.t. a filtration $(\mathcal F_t, t\ge 0)$ is a process $(X_t, t\ge 0)$ that can be written as $X_t = M_t+V_t$ where
- $(M_t, t\ge 0)$ is a continuous square-integrable martingale w.r.t. $(\mathcal F_t, t\ge 0)$.
- $(V_t, t \ge 0)$ is a continuous process that has bounded variation and is adapted to $(\mathcal F_t, t\ge 0)$ such that $V_0 = 0$ a.s.
Example 1. Let $H$ and $K$ be continuous adapted processes such that $\mathbb E [\int_0^t H_s^2 \mathrm d s] <\ \infty$ for all $t\ge 0$. Then the process $X = (X_t, t\ge 0)$ defined as $$ X_t := \underbrace{X_0 + \int_0^t H_s \mathrm d B_s}_{M_t} + \underbrace{\int_0^t K_s \mathrm d s}_{V_t} \quad \forall t\ge 0. $$ is a continuous semi-martingale.
Example 2. Let $(B_t, t \ge 0)$ be the standard Brownian motion and $f:\mathbb R \to \mathbb R$ twice continuously differentiable such that $\mathbb E [\int_0^t (f'(B_s))^2 \mathrm d s ] < \infty$. By Itô's lemma, $$ f(B_t) = \underbrace{f(B_0) + \int_0^t f'(B_s) \mathrm d B_s}_{M'_t} + \underbrace{\frac{1}{2} \int_0^t f''(B_s) \mathrm d s}_{V'_t} \quad \text{a.s.} \quad \forall t \ge 0. $$ Then $(f(B_t), t \ge 0)$ is a continuous semi-martingale.
In Example 2, $f(B_t) = M'_t+V'_t$ a.s. So for each $t \ge 0$, $f(B_t)$ and $M_t+V_t$ can differ on a $\mathbb P$-null set. I am not sure if there is $(M_t, V_t)$ with required properties such that $M_t (\omega) + V_t (\omega) = f(B_t) (\omega)$ for all $\omega \in \Omega$. The definition mentioned in this Wikipedia page also uses everywhere inequality in the decomposition $X_t = M_t+V_t$, i.e., the phrase "a.s." does not appear.
Could you elaborate on my confusion?