Show the solution to Poisson's equation in $\mathbb{R}^2$ is well-defined

Question

In Mathematical Theory of Incompressible Nonviscous Fluids by Marchioro and Pulvirenti, the authors define a function $$ \Phi(x) = -\frac{1}{2\pi}\int_{\mathbb{R}^2} \log|x-y|\omega(y)\ dy$$ (the solution to Poisson's equation $-\Delta \Phi = \omega$ in $\mathbb{R}^2$) and assert that the function definition will make sense if $\omega \in L^1(\mathbb{R}^2) \cap L^\infty(\mathbb{R}^2)$. Now I can show that the integral is defined over any bounded set (including on bounded sets containing the singularity of the logarithm), but how do we show the integral is defined on all of $\mathbb{R}^2$? To clarify, I don't need to show $\Phi$ in fact solves Poisson's equation, but that the integrand will be in $L^1(\mathbb{R}^2)$ for any choice of $x$.