Making use of completion relation to find general expectation

[ChemicalTom]

I am stuck on this Self-test 1.6 in molecular quantum mechanics by atkins and friedman.
Probably making use of the completeness relation the question is the following: Show that if <Ωf>*=-Ωf*, then <Ω>=0 for any real function f.
Anyone got a clue?

 

[Asim Riaz]

A:Let's see what we can do.Using the completeness relation, we can write$$\langle \Omega \rangle = \int_{-\infty}^{\infty}\Omega(x)f(x) dx = \int_{-\infty}^{\infty}\Omega^*(x)f(x) dx = \langle \Omega^* \rangle$$Since $\langle \Omega f \rangle = -\langle \Omega^* f \rangle$ and $f$ is real, we have$$\langle \Omega \rangle = \langle \Omega^* \rangle = -\langle \Omega^* \rangle = -\langle \Omega \rangle \implies \langle \Omega \rangle = 0$$