Q:

Bayesian Estimation: calculating an integral

I am reading a book on Bayesian filtering and I have a question regarding calculating transition density $p(X_t|X_{t-1})$. 
My question is how the term $p(X_t|X_{t-1}, V_{t}=v)$ is converted to the Dirac delta function $\delta(X_t - f(X_{t-1}, v))$ ?
Please see the image below.

Another question is, if we have a discrete-time nonlinear state equation with additive noise: $X_t = f(X_{t-1}) + V_t$, now, how we can compute the following PDFs:
1- $P(X_t)$ 
2- $P(X_t|X_{t-1}, V_t)$
Suppose that the distribution of the noise $p (V_t)$ is known.
Note that it is obvious that the $P(X_t|X_{t-1})$ is computed according to the total probability law.

A:

To answer your first question, note that $X_t$ depends only on $X_{t-1}$ and $V_t$ through the evolution equation $X_t = f(X_{t-1}, V_t).$ Hence given that we know the values of $X_{t-1}$ and $V_t,$ $X_t$ is not random anymore and hence its pdf is given by a Dirac delta function. Hence $X_t$ occurs with probability one at $f(X_{t-1}, V_t)$ (the Dirac delta function represents the pdf of a degenerate random variable).
To answer the second question, first part
 \begin{align}
p(X_t = x_t) &= \int_{-\infty}^{\infty} p(x_t | x_{t-1}, v_t)\, p(x_{t-1} | v_t)\, p(v_t) \, dx_{t-1} \, dv_t \\
&= \int_{-\infty}^{\infty} \delta(x_t - (f(x_{t-1}) + v_t)) \, p(x_{t-1}) \, p(v_t) \, dx_{t-1} \, dv_t \\
& = \int_{\{  (x_{t-1}, v_t) : f(x_{t-1}) + v_t = x_t\}} p(x_{t-1}) \, p(v_t) \, dx_{t-1} \, dv_t.
\end{align}
The answer to the second part is again the delta function $\delta(X_t - (f(X_{t-1}) + V_t)).$

