Is $(f\mapsto\int_\Omega f d\mu)\in L^1(\Omega,\lambda)^*$?

Consider any point $x_0 \in \Omega$ and let $\delta_{x_0}$ be the Dirac measure, i.e. $\delta_{x_0}(A) = 1(x_0 \in A)$ for any $A \in \mathfrak{B}(\Omega)$. This is a Borel regular measure. Define the function $f(x) = 1(x = x_0)$, so $f(x) = 1$ for $x = x_0$ and $f(x) = 0$ otherwise. Then $$ \int f d\delta_{x_0} = 1 $$ but $$ \int |f| d\lambda = 0. $$

You don't need to decide whether the integral of $f$ is finite or not. If $\int_\Omega f~\mathrm d\mu$ is not finite, then $f$ is not $L^1$ ! Now that we know all these integrals are finite, the linearity of the integral shows this map is linear, and hence is a member of the dual space of $L^1(\Omega,\mathfrak{B}(\Omega),\mu)$.