Multiplicativity of expectation for real functions implies the same for complex-valued

Question

Suppose that $$\mathbb{E} [f_1(X_1) \ldots f_M(X_M)] = \mathbb{E}[f_1(X_1)]\ldots\mathbb{E}[f_M(X_M)]$$ for all functions $f: \mathbb{R} \rightarrow \mathbb{R}$. Is it necessarily true that this also holds for all functions $g: \mathbb{R} \rightarrow \mathbb{C}$? Just personally curious, I can't seem to prove it though.

Answer 1

Yes, use linearity of expectation after expanding out the complex-valued function into its real and imaginary parts.

Let $h_m(x):=f_m(x)+ig_m(x)$ for real-valued $f_m$ and $g_m$. \begin{align*} \mathbb{E}[h_1(X_1) h_2(X_2)] &=\mathbb{E}[((f_1(X_1)+ig_1(X_1))(f_2(X_2)+ig_2(X_2))]\\ &=\mathbb{E}[f_1(X_1)f_2(X_2)]-\mathbb{E}[g_1(X_1)g_2(X_2)]+i\left(\mathbb{E}[g_1(X_1)f_2(X_2)]+\mathbb{E}[f_1(X_1)g_2(X_2)]\right)\\ &=\mathbb{E}[f_1(X_1)]\mathbb{E}[f_2(X_2)]-\mathbb{E}[g_1(X_1)]\mathbb{E}[g_2(X_2)]+i\left(\mathbb{E}[g_1(X_1)]\mathbb{E}[f_2(X_2)]+\mathbb{E}[f_1(X_1)]\mathbb{E}[g_2(X_2)]\right)\\ &=\mathbb{E}[h_1(X_1)]\mathbb{E}[h_2(X_2)]. \end{align*}