Suppose that $G$ be a locally compact abelian group. Let $I$ be a right translation invariant subspace of $L^1(G)$; that means $f_x$ is in $I$ for all $f$ in $I$ and $x$ in $G$ and let $f_x(y):=f(y-x)$ and also suppose $\phi$ be in $L^∞(G)$ that annihilates $I$; that means $\phi(g)=0$, for $g\in I$. Then by [1] $$\int_Gf(−y)ϕ(y)dy=0$$ for every $f\in I$
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Question: Why does this equation is hold? or Is this equation valid for any locally compact group?
[1] Walter Rudin, Fourier analysis on groups,1962,page 157.