How to derive the distribution of measurement noise in discrete Kalman filter which is transformed from continuous one? With sampling time $T$, and a continuous measuring model:
$$
\begin{align}
y(t) &= Cx(t)+v(t) \\
v(t) & \sim \text{N}(0,R_c)
\end{align}
$$
we can change it into a practical discrete one, which is 
$$
\begin{align}
y_k &= Cx_k+v_k \\
v_k & \sim \text{N}(0,R)
\end{align}
$$
and
$$
\mathbf{R=\frac{R_c}{T}} \tag{$*$}\label{a}
$$
how to get the $\eqref{a}$ ?
I don't quit understand this because for a sampling process, we say
$$
v_k=v(kT)=v(t)
$$
so $v_k$ is as same as $v(t)$ in a particular time $t$. so $v_k$ and $v(t)$ have the same distribution, leading $$\text{Cov}(v_k)=\text{Cov}(v(t))$$
that means $R=R_c$.
Or using equation $\eqref{a}$ to promise the discrte Kalman filter to have the same conclusion with the continuous one?
EDIT: equation $\eqref{a}$ is used in the MIT lecture.
 A: After a lot of digging the last few days and coming across this question on my way, I figured I'd assist future Googlers. Applied Optimal Estimation by Gelb (ISBN 0-262-57048-3), section 4.3 Continuous Kalman Filter, page 121 explains how to discretize the continuous-time measurement noise covariance matrix. To paraphrase it:
$\textbf{R}(t)$ is continuous and $\textbf{R}_k$ is discrete here. Consider the discrete white noise sequence $\textbf{v}_k$ and the (non-physically realizable) continuous white noise process $\textbf{v}$. Whereas $\textbf{R}_k = E[\textbf{v}_k \textbf{v}_k^T]$ is a covariance matrix, $\textbf{R}(t)$ defined by $E[\textbf{v}(t) \textbf{v}^T(\tau)] = \textbf{R}(t)\delta(t - \tau)$ is a spectral density matrix (the Dirac function $\delta(t - \tau)$ has units of $1/\text{sec}$). The covariance matrix $\textbf{R}(t)\delta(t - \tau)$ has infinite-valued elements. The discrete white noise sequence can be made to approximate the continuous white noise process by shrinking the pulse lengths ($T$) and increasing their amplitude, such that $\textbf{R}_k \rightarrow \frac{1}{T}\textbf{R}$.
That is, in the limit as $T \rightarrow 0$, the discrete noise sequence tends to one of infinite-valued pulses of zero duration such that the area under the "impulse" autocorrelation function is $\textbf{R}_k T$. This is equal to the area $\textbf{R}$ under the continuous white noise impulse autocorrelation function.
