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

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

Hint: If $$f,g\in L^1(G)$$ it's easy to verify from the definition that $$\int_Gf*g=\int_G f\int_Gg.$$
• @user62498 Come on now! Say $I$ is that set. It should be obvious that $I$ is the kernel of a bounded linear functional. Or if that's too abstract, assume $f_n\in I$, $f_n\to f$ and show $f\in I$. – David C. Ullrich Apr 29 at 16:32