Integrating holomorphic L¹-functions

Let $\Omega\Subset\mathbb{C}^{n}$ be a domain with smooth boundary, $k\in\mathbb{N}$. It is shown that the integral of a holomorphic function in $L^1(\Omega)$ may be represented as the integral of this function against a smooth function vanishing to order $k-1$ on $b\Omega$. An application for a smoothing property of the Bergman projection for conjugate holomorphic functions is given.