[proofplan] We first separate the prediction error into the test noise $\varepsilon_0$ and the estimation error $f_0(X_0)-\hat f_S(X_0)$. The mixed term vanishes because the training sample is independent of the test observation and the test noise has conditional mean zero given $X_0$. We then condition on $X_0$ and decompose the remaining squared estimation error into its conditional mean square and conditional variance. Taking expectations gives the irreducible noise, squared bias, and variance terms. [/proofplan]

[step:Split the prediction error into noise and estimation terms]Define the real-valued random variable \begin{align*} Z:=\hat f_S(X_0). \end{align*} By the model relation $Y_0=f_0(X_0)+\varepsilon_0$, we have \begin{align*} Y_0-Z=\varepsilon_0+\bigl(f_0(X_0)-Z\bigr). \end{align*} Since all terms are square-integrible, expanding the square and taking expectations gives \begin{align*} \mathbb E[(Y_0-Z)^2] &= \mathbb E[\varepsilon_0^2] +2\,\mathbb E\left[\varepsilon_0\bigl(f_0(X_0)-Z\bigr)\right] +\mathbb E\left[\bigl(f_0(X_0)-Z\bigr)^2\right]. \end{align*}[/step]

[guided]Set \begin{align*} Z:=\hat f_S(X_0). \end{align*} This notation isolates the random prediction at the test point. The randomness in $Z$ comes from both the random test feature $X_0$ and the independent training sample $S$. Using the identity $Y_0=f_0(X_0)+\varepsilon_0$, the prediction error is \begin{align*} Y_0-Z=\varepsilon_0+\bigl(f_0(X_0)-Z\bigr). \end{align*} The first term is the new test noise, while the second term is the error made in estimating the regression function value $f_0(X_0)$. Because $Y_0$, $f_0(X_0)$, and $Z$ are square-integrable, every term in the square expansion is integrable. Therefore \begin{align*} \mathbb E[(Y_0-Z)^2] &= \mathbb E[\varepsilon_0^2] +2\,\mathbb E\left[\varepsilon_0\bigl(f_0(X_0)-Z\bigr)\right] +\mathbb E\left[\bigl(f_0(X_0)-Z\bigr)^2\right]. \end{align*} The rest of the proof identifies these three terms.[/guided]

[step:Show that the mixed noise and estimation term vanishes]Since $S$ is independent of $(X_0,Y_0)$ and $\varepsilon_0=Y_0-f_0(X_0)$ is measurable with respect to $\sigma(X_0,Y_0)$, independence gives \begin{align*} \mathbb E[\varepsilon_0\mid X_0,S] = \mathbb E[\varepsilon_0\mid X_0] = 0 \end{align*} almost surely. The random variable $f_0(X_0)-Z$ is measurable with respect to $\sigma(X_0,S)$, so the defining property of conditional expectation gives \begin{align*} \mathbb E\left[\varepsilon_0\bigl(f_0(X_0)-Z\bigr)\right] &= \mathbb E\left[ \bigl(f_0(X_0)-Z\bigr)\mathbb E[\varepsilon_0\mid X_0,S] \right] \\ &=0. \end{align*}[/step]

[guided]We need to prove that the cross term contributes nothing. The factor $f_0(X_0)-Z$ is determined once $X_0$ and the training sample $S$ are known, so it is measurable with respect to $\sigma(X_0,S)$. The noise $\varepsilon_0=Y_0-f_0(X_0)$ is measurable with respect to $\sigma(X_0,Y_0)$. Since $S$ is independent of $(X_0,Y_0)$, conditioning additionally on $S$ does not change the conditional mean of $\varepsilon_0$ once $X_0$ is known. Hence \begin{align*} \mathbb E[\varepsilon_0\mid X_0,S] = \mathbb E[\varepsilon_0\mid X_0] = 0 \end{align*} almost surely. Now use the defining property of conditional expectation with the $\sigma(X_0,S)$-measurable multiplier $f_0(X_0)-Z$: \begin{align*} \mathbb E\left[\varepsilon_0\bigl(f_0(X_0)-Z\bigr)\right] &= \mathbb E\left[ \bigl(f_0(X_0)-Z\bigr)\mathbb E[\varepsilon_0\mid X_0,S] \right] \\ &=0. \end{align*} Thus the test noise is orthogonal, in expectation, to the estimation error.[/guided]

[step:Identify the irreducible noise term] Because $\mathbb E[\varepsilon_0\mid X_0]=0$, the conditional variance assumption gives \begin{align*} \mathbb E[\varepsilon_0^2\mid X_0] = \operatorname{Var}(\varepsilon_0\mid X_0) = \sigma^2 \end{align*} almost surely. Taking expectations yields \begin{align*} \mathbb E[\varepsilon_0^2]=\sigma^2. \end{align*} [/step]

[step:Decompose the conditional estimation error into bias and variance]Define the conditional mean prediction \begin{align*} m:\Omega&\to\mathbb R,\\ \omega&\mapsto \mathbb E[Z\mid X_0](\omega). \end{align*} Then $m$ is $\sigma(X_0)$-measurable. Write \begin{align*} f_0(X_0)-Z = \bigl(f_0(X_0)-m\bigr)+\bigl(m-Z\bigr). \end{align*} Conditioning on $X_0$ and expanding the square gives \begin{align*} \mathbb E\left[\bigl(f_0(X_0)-Z\bigr)^2\mid X_0\right] &= \bigl(f_0(X_0)-m\bigr)^2 +2\bigl(f_0(X_0)-m\bigr)\mathbb E[m-Z\mid X_0] \\ &\quad +\mathbb E\left[(m-Z)^2\mid X_0\right]. \end{align*} Since $m=\mathbb E[Z\mid X_0]$, we have \begin{align*} \mathbb E[m-Z\mid X_0]=m-\mathbb E[Z\mid X_0]=0. \end{align*} Also, by the definition of conditional variance, \begin{align*} \mathbb E\left[(m-Z)^2\mid X_0\right] = \operatorname{Var}(Z\mid X_0). \end{align*} Therefore \begin{align*} \mathbb E\left[\bigl(f_0(X_0)-Z\bigr)^2\mid X_0\right] = \bigl(f_0(X_0)-\mathbb E[Z\mid X_0]\bigr)^2 + \operatorname{Var}(Z\mid X_0). \end{align*}[/step]

[guided]The remaining term is the squared error of the fitted rule around the true regression function value. We split it into a systematic part and a centered random part. Define \begin{align*} m:\Omega&\to\mathbb R,\\ \omega&\mapsto \mathbb E[Z\mid X_0](\omega). \end{align*} The random variable $m$ is the average prediction at the observed test feature $X_0$, where the averaging is over the training randomness conditional on $X_0$. It is $\sigma(X_0)$-measurable by definition of conditional expectation. Now decompose \begin{align*} f_0(X_0)-Z = \bigl(f_0(X_0)-m\bigr)+\bigl(m-Z\bigr). \end{align*} The first term is the conditional bias at $X_0$. The second term is centered conditional on $X_0$, because \begin{align*} \mathbb E[m-Z\mid X_0] = m-\mathbb E[Z\mid X_0] = 0. \end{align*} Expanding the conditional square gives \begin{align*} \mathbb E\left[\bigl(f_0(X_0)-Z\bigr)^2\mid X_0\right] &= \bigl(f_0(X_0)-m\bigr)^2 +2\bigl(f_0(X_0)-m\bigr)\mathbb E[m-Z\mid X_0] \\ &\quad +\mathbb E\left[(m-Z)^2\mid X_0\right]. \end{align*} The middle term is zero by the centering just proved. The final term is exactly the conditional variance of $Z$ given $X_0$, since $m=\mathbb E[Z\mid X_0]$: \begin{align*} \mathbb E\left[(m-Z)^2\mid X_0\right] = \operatorname{Var}(Z\mid X_0). \end{align*} Thus \begin{align*} \mathbb E\left[\bigl(f_0(X_0)-Z\bigr)^2\mid X_0\right] = \bigl(f_0(X_0)-\mathbb E[Z\mid X_0]\bigr)^2 + \operatorname{Var}(Z\mid X_0). \end{align*}[/guided]

[step:Take expectations to obtain the decomposition] Taking expectations in the conditional identity from the previous step gives \begin{align*} \mathbb E\left[\bigl(f_0(X_0)-Z\bigr)^2\right] = \mathbb E\left[\bigl(f_0(X_0)-\mathbb E[Z\mid X_0]\bigr)^2\right] + \mathbb E\left[\operatorname{Var}(Z\mid X_0)\right]. \end{align*} Combining this identity with the expansion of $\mathbb E[(Y_0-Z)^2]$, the vanishing mixed term, and $\mathbb E[\varepsilon_0^2]=\sigma^2$, we obtain \begin{align*} \mathbb E\left[(Y_0-\hat f_S(X_0))^2\right] &= \sigma^2 +\mathbb E\left[\left(f_0(X_0)-\mathbb E[\hat f_S(X_0)\mid X_0]\right)^2\right] \\ &\quad +\mathbb E\left[\operatorname{Var}(\hat f_S(X_0)\mid X_0)\right]. \end{align*} This is the claimed bias–variance decomposition. [/step]