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We study the density of polynomials in $H^2(E,\varphi)$, the space of squareintegrable functions with respect to $e^{-\varphi}dm$ and holomorphic on theinterior of $E$ in $\mathbb{C}$, where $\varphi$ is a subharmonic function and$dm$ is a measure on $E$. We give a result where $E$ is the union of aLipschitz graph and a Carath\'{e}odory domain, that we state as a weighted$L^2$-version of the Mergelyan theorem. We also prove a weighted $L^2$-versionof the Carleman theorem.  Keywords: Mergelyan theorem, Carleman theorem, Weighted $L^2$- spaces,Rectifiable non-Lipschitz arc

Weighted $L^2$ version of Mergelyan and Carleman approximation