 Solution

Write ${\sigma }^2={\Sigma }_{\eta }>0$ and assume $V_0>0$. The scalar Kalman filter reduces toTaking reciprocal variance and dividing the mean by variance givesThusSince the process noise is zero and $A=1$, $X_t=X_0$ and $Y_j=X_0+{\eta }_j$. The strong law of large numbers gives $t^{-1}{\sum }_j{Y}_j\to X_0$ almost surely. The prior's finite weight vanishes in the displayed formula, so both limits equal $X_0$ almost surely. A zero prior variance instead means $X_0$ is known, and the assertion follows directly without reciprocal-variance formulas.