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In Lars Hormander's An Introduction to Complex Analysis in Several Variables, page 77, he says that"every function in $L^2(\Omega, loc)$ belongs to $L^2(\Omega,\phi)$ for some continuous $\phi$ , where $L^2(\Omega,\phi)$ has the measure $e^{-\phi}d\lambda$. Why is that?

$\Omega$is open and $d\lambda$ is the Lesbegue measure

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    If you write $\Omega = \cup_{n=1}^{\infty} K_n$ where $K_n$ are a nested sequence of compact sets, could you define the appropriate $\phi$ "piecewise" on each $K_{n+1} \setminus K_n$ in a continuous way? This could be one way to do this. – 2017-02-01
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    I do this in another way, by ajudusting $\phi$ to the $L^2$ norm of $f$. – 2017-02-01

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