4. Given a differentiable function $f(x)(x \in \mathbf{R})$ that satisfies $f^{\prime}(x)>f(x)$. Then when $a>0$, the relationship between $f(a)$ and $\mathrm{e}^{a} f(0)$ is ( ).
(A) $f(a)\mathrm{e}^{a} f(0)$
(C) $f(a)=e^{a} f(0)$
(D) Cannot be determined, depends on $f(x)$ or $a$
4.B.

Let the function $F(x)=\frac{f(x)}{\mathrm{e}^{x}}(x \in \mathbf{R})$, then $F^{\prime}(x)=\frac{f^{\prime}(x)-f(x)}{\mathbf{e}^{x}}>0$.
Therefore, $F(x)$ is a monotonically increasing function on $\mathbf{R}$. Thus, for $a>0$, we have
$$
F(a)>F(0) \Rightarrow \frac{f(a)}{\mathrm{e}^{a}}>\frac{f(0)}{\mathrm{e}^{0}},
$$

which implies $f(a)>\mathrm{e}^{a} f(0)$.