Closed ideal in $ L^{1}(G)$

Question

Let $G$ be locally compact group prove that

$$L_{0}^{1}(G)=\left\{f\in L^{1}(G): \int_G f(g) dm(g)=0 \right\}$$ is a closed ideal in $ L^{1}(G)$ with codimension one

I am grateful for any help

Answer 1

Hint: If $f,g\in L^1(G)$ it's easy to verify from the definition that $$\int_Gf*g=\int_G f\int_Gg.$$