  We introduce a novel notion of {\it local spectral gap} for general, possibly
infinite, measure preserving actions. We establish local spectral gap for the
left translation action $\Gamma\curvearrowright G$, whenever $\Gamma$ is a
dense subgroup generated by algebraic elements of an arbitrary connected simple
Lie group $G$. This extends to the non-compact setting recent works of Bourgain
and Gamburd \cite{BG06,BG10}, and Benoist and de Saxc\'{e} \cite{BdS14}. We
present several applications to the Banach-Ruziewicz problem, orbit equivalence
rigidity, continuous and monotone expanders, and bounded random walks on $G$.
In particular, we prove that, up to a multiplicative constant, the Haar measure
is the unique $\Gamma$-invariant finitely additive measure defined on all
bounded measurable subsets of $G$.
