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Yes, $f=x$ almost everywhere wrt $P$. Consider the push-forward measure $\mu=f_*P$ (in probability, it is also called the distribution of $f$). It is defined as $\mu(E)=P(f^{-1}(E))$. This $\mu$ is a probability measure on $C$. Your condition becomes $$\int_C d\mu=x,$$ and this implies that $\mu$ is an atom at $x$.