Weinberg Volume II: Abelian Anomaly Function The following is from page 363 of Weinberg volume II.
We wish to evaluate the RHS of 
\begin{align}\label{EQbbvbv}
[d \psi][d \bar{\psi}] \rightarrow(\operatorname{Det} \mathscr{U} \operatorname{Det} \overline{\mathscr{U}})^{-1}[d \psi][d \bar{\psi}],
\end{align}
in order to find the effect on the measure of a change of path integration variables corresponding to a local matrix transformation $\begin{align}
\psi(x) \rightarrow U(x) \psi(x)
\end{align}$. We define 
\begin{align}\notag
\mathscr{U}_{x n, y m} &:=U(x)_{n m} \delta^{4}(x-y), \text{ and}\\\label{EQdwdf22vfr}
%%
%%
\bar{\mathscr{U}}_{x n, y m} &:=\left[\gamma_{4} U(x)^{\dagger} \gamma_{4}\right]_{n m} \delta^{4}(x-y),
\end{align}
where $\gamma_{4} := i \gamma^{0}$ is used in defining
$\bar{\psi}=\psi^{\dagger} \gamma_{4}$.
Also note that the indices $n,m$ run over flavour labels and Dirac spin indices. 
Let us consider $\alpha(x)$ to be an infinitesimal scalar function in the 
transformation
\begin{align}\label{EQmnmmb4}
U(x)=\exp \left[i \gamma_{5} \alpha(x) t\right]
\end{align}
where $t$ is a general hermitian matrix. 
Note that Weinberg omits the working out from  here, so the following 3 equations are my own work. 
In this case, since the Taylor expansion of the exponential has negligible contributions from terms of order greater the one in $\alpha$, one obtains that
\begin{align}
[\mathscr{U}-1]_{n x, m y}=i \alpha(x)[\gamma _5 t]_{n m} \delta^{4}(x-y).
\end{align}
Therefore,
\begin{align}
\operatorname{Det} \mathscr{U} &= \exp{\text{Tr} \ln \{1+i \alpha(x)\left[\gamma_{5} t\right]_{n m} \delta^{4}(x-y)\}}\\
%%
%%
&= \exp{\, i \alpha(x)\text{Tr}\{\gamma_{5} t\} \delta^{4}(x-y)},
\end{align}
where we have used the identity for the determinant of a matrix $M$, $\operatorname{Det} M=\exp \operatorname{Tr} \ln M$, and that $\ln (1+x)\rightarrow x$ as $x\rightarrow 0.$ But since $\mathscr U$ is pseudo-Hermitian,
$(\begin{align}
\overline{\mathscr{U}}=\mathscr{U}
\end{align})$
we have
\begin{align}
[d \psi][d \bar{\psi}] \rightarrow(\operatorname{Det} \mathscr{U})^{-2}[d \psi][d \bar{\psi}].
\end{align}
Weinberg now claims that the measure changes under this transformation as
\begin{align}\label{EQnmnnghr3}
[d \psi][d \bar{\psi}] \rightarrow \exp \left\{i \int d^{4} x \alpha(x) \mathscr{A}(x)\right\}[d \psi][d \bar{\psi}],
\end{align}
where we define the anomaly function
\begin{align}\label{EQvccvezz33r43}
\mathscr{A}(x)=-2 \operatorname{Tr}\left\{\gamma_{5} t\right\} \delta^{4}(x-x).
\end{align}
We use `Tr' to denote a trace to be taken over Dirac and species indices.
Question 1: Where does the $\delta^{4}(x-x)$ comes from? I can't see any reason why the argument of the delta function might change in the calculations I went through.
Question 2: Where does the integral over $d^4 x$ comes from in the second last equation? If we're working with the Jacobian as on the RHS of the first equation, I don't see how an integral could pop up. 
Question 3: This is definitely a rather trivial question, but I've never encountered $\gamma^4$ before... I'm used to the definition $\bar{\psi}=\psi^{\dagger} \gamma_{0}$. This is probably just a different representation of the spinors. If so, could I get a name for it please? I can't seem to find it. Final trivial question: what is a species index? 
 A: Questions 1. and 2. are intimately related. 
To try to make sense out of the $\det = \exp \rm Tr \ln$ formula, the meaning of the trace has to be elaborated, I quote QFT and the Standard Model by Schwartz, line 30.58:
$$\det U = \exp \rm Tr \log U = \exp \int d^4x \langle{x}|\rm tr \ln U |x \rangle{}$$
Meaning that the $\rm Tr$ written with uppercase letters here has a double meaning - Tracing over the internal indices (spelled out $n,m$) in your question, and represented by the lowercase $\rm tr$ in the above formula, and "tracing" over the "spacetime indices", represented by the spacetime integral, and by using a one particle Hilbert space here only as a mathematical trick to be able to perform the tracing out.
So in the example above you need the trace of $(\gamma_{5} t) \mathbf{I} $ where the first part $(\gamma_{5} t)$ carries fermionic/color indices and $\mathbf{I}$ is an identity operator with "spacetime" indices, which is above represented by $\delta^{(4)} (x-y)$. Thus 
$$\rm Tr\{\gamma_{5} t \mathbf{I} \} = \int d^4 x \langle{x}|\rm tr (\gamma_{5} t) \mathbf{I} |x \rangle{} \\ = \rm tr (\gamma_{5} t) \int d^4x \langle{x}|x \rangle{}  \\ = \int d^4x \rm tr (\gamma_{5} t) \delta^{(4)}(x-x)$$
For Question 3, I think this amounts only to a specific notation shown at the beggining of Weinberg's textbook.
