For a compact Lie group, $G$, let ${\cal L}(G)$ be the free loop group consisting of the space of continuous paths $g:[0,1]\rightarrow G$ with $g(0)=g(1)$. For a given $Ad_{G}$-invariant inner product on the Lie algebra of $G$ and a suitable related inner product on the Lie algebra of ${\cal L}% (G)$ we derive the corresponding Riemannian structure on ${\cal L}(G)$. For this structure we have constructed finite dimensional approximations which are used in proving a logarithmic Sobolev inequality and integration by parts formulas on ${\cal L}(G)$. The underlying probability measure on ${\cal L}(G)$ can be derived by an ${\cal L}(G)$-valued Brownian motion.