How does the functional measure transform under a field redefinition? My question is: how does the path integral functional measure transform under the following field redefinitions (where $c$ is an arbitrary constant and $\phi$ is a scalar field):
\begin{align}
\phi(x)&=\theta(x)+c \,\theta^3(x) \tag{1}\\
\phi(x)&=c\,\theta^3(x) \tag{2}\\
\phi(x)&=\sinh\big(\theta(x)\big)\tag{3}
\end{align}
My naive guess for the transformation in Eq.(3) is
\begin{align}
\mathcal{D}\phi&=\mathcal{D}\theta\,\,\text{Det}\Bigg[\frac{\delta \phi(x)}{\delta\theta(x')}\Bigg]=\mathcal{D}\theta \,\,\text{Det}\bigg[\cosh(\theta(x))\delta(x-x')\bigg]\\
&=\mathcal{D}\theta\exp\bigg[\text{Tr}\,\Big(\log\big(\cosh(\theta(x))\delta(x-x')\big)\Big)\bigg]\\
&=\mathcal{D}\theta\exp\bigg[\int dx\,\,\log\Big(\cosh(\theta(x))\delta(x-x')\Big)\bigg]
\end{align}
But that seems very wrong.
 A: *

*It seems natural to generalize OP's setting to several fields $\phi^{\alpha}$ in $d$ spacetime dimensions. Consider for simplicity ultra-local field redefinitions$^1$
$$\begin{align} \phi^{\prime\alpha}(x)~=~&F^{\alpha}(\phi(x),x)\cr
~=~&\phi^{\alpha}(x)-f^{\alpha}(\phi(x),x).\end{align}\tag{1} $$


*The Jacobian functional determinant in the path/functional integral is formally given as
as a functional Berezin integral
$$ \begin{align} J~=~&{\rm Det} (\mathbb{M})\cr
~=~&\int \!{\cal D}c~{\cal D}\bar{c} \exp\left(\int\! d^dx^{\prime} \!\int\! d^dx ~\bar{c}_{\beta}(x^{\prime})\mathbb{M}^{\beta}{}_{\alpha}(x^{\prime},x)c^{\alpha}(x)  \right)
,\end{align} \tag{2}$$
where $c^{\alpha}(x)$ and $\bar{c}_{\beta}(x^{\prime})$ are Grassmann-odd ghost fields;
or as
$$\begin{align} J~=~&{\rm Det} (\mathbb{M})\cr
~=~&\exp {\rm Tr}\ln (\mathbb{M})\cr
~=~& \exp\left(-\sum_{j=1}^{\infty} \frac{1}{j}{\rm Tr} (\mathbb{m}^j)\right) \cr
~=~& \exp\left(\delta^d(0) \int\! d^dx ~{\rm tr} (\ln M(x))\right),\end{align} \tag{3} $$
where we have defined
$$\begin{align}  \mathbb{M}~\equiv~&\mathbb{1}-\mathbb{m},\cr \mathbb{M}^{\beta}{}_{\alpha}(x^{\prime},x) 
~:=~&\frac{\delta F^{\beta}(x^{\prime})}{\delta\phi^{\alpha}(x)}\cr
~=~& M^{\beta}{}_{\alpha}(x^{\prime})\delta^d(x^{\prime}\!-\!x),\cr 
M^{\beta}{}_{\alpha}(x)~:=~& \frac{\partial F^{\beta}(x)}{\partial\phi^{\alpha}(x)}~=~\delta^{\beta}_{\alpha}-m^{\beta}{}_{\alpha}(x),\cr 
\mathbb{m}^{\beta}{}_{\alpha}(x^{\prime},x)
~:=~&\frac{\delta f^{\beta}(x^{\prime})}{\delta\phi^{\alpha}(x)}\cr
~=~& m^{\beta}{}_{\alpha}(x^{\prime})\delta^d(x^{\prime}\!-\!x),\cr 
m^{\beta}{}_{\alpha}(x)~:=~& \frac{\partial f^{\beta}(x)}{\partial\phi^{\alpha}(x)}.\end{align} \tag{4}$$


*A local field redefinition corresponds to insertion of UV-relevant/IR-irrelevant terms in the action, i.e. it doesn't change low-energy physics.


*If we discretize spacetime, then the Jacobian (3) becomes a product of ordinary determinants
$$ J~=~\prod_i \det (M(x_i)), \tag{5}$$
where the index $i$ labels lattice points $x_i$ of spacetime. The Dirac delta at zero $\delta^d(0)$ is here replaced by a reciprocal volume of a unit cell of the spacetime lattice, which can viewed as a UV regulator, cf. e.g. my Phys.SE answer here.


*In dimensional regularization (DR), the Jacobian $J=1$ becomes one under local field redefinitions (if there are no anomalies present).

*

*A ghost-loop diagram in momentum space is UV-divergent (because the ghost propagator is $1$ rather than $\frac{1}{k^2}$, so there are no negative powers of momenta in the diagram), cf. Refs. 1-7. The scaleless integral is then proportional to a positive power of an IR-mass regulator, and hence zero when the IR-mass regulator is removed.


*The Dirac delta at zero $$\delta^d(0)~=~\int \!\frac{d^dk}{(2\pi)^d} (k^2)^0~=~0 \tag{6}$$  vanishes, cf. Refs. 1-7. Heuristically, DR only picks up residues of various finite parameters of the physical system, while contributions from infinite parameters are regularized to zero.
References:

*

*G 't Hooft & M.J.G. Veltman, Diagrammar, CERN report, 1973; p. 46-51.


*G. Leibbrandt, Introduction to the technique of dimensional regularization, Rev. Mod. Phys. 47 (1975) 849; Subsection IV.B.3 p. 864.


*M. Henneaux & C. Teitelboim, Quantization of Gauge Systems, 1994; Subsection 18.2.4.


*I.V. Tyutin, Once again on the equivalence theorem, arXiv:hep-th/0001050
.


*R. Ferrari, M. Picariello & A. Quadri, An Approach to the Equivalence Theorem by the Slavnov-Taylor Identities, arXiv:hep-th/0203200.


*A.V. Manohar, Introduction to Effective Field Theories, arXiv:1804.05863; p. 33-34 & p. 51.


*J.C. Criado & M. Perez-Victoria, Field redefinitions in effective theories at higher orders, arXiv:1811.09413.
--
$^1$ Much of this can be generalized to local field redefinitions
$$ \begin{align}\phi^{\prime\alpha}(x)~=~&F^{\alpha}(\phi(x),\partial\phi(x),\partial^2\phi(x), \ldots ,\partial^N\phi(x) ,x)\cr
~=~&\phi^{\alpha}(x)\cr
~-~&f^{\alpha}(\phi(x),\partial\phi(x),\partial^2\phi(x), \ldots ,\partial^N\phi(x) ,x),\end{align}\tag{7}$$
and to a certain degree even beyond; and to derivatives $\partial^j\delta^d(0)$ of the Dirac delta at zero, cf. Refs. 1-7.
A: All three cases, (1)-(3), are local redefinitions, meaning that the value of $\phi(x)$ for any given $x$ is determined only by the value of $\theta(x)$ at that same value of $x$ (and conversely, assuming it's invertible). 
Conceptually, the parameter $x$ is just a continuous index labeling different integration variables. In fact, the most generally-applicable way we have for defining a functional integral (at least in QFT) is to replace this continuous parameter with a discrete index. Then you have an ordinary multi-variable integral, and the rule for changing integration variables is the usual one. So the cases (1)-(3) just describe changes-of-variable in a bunch of single-variable integrals.
Thinking about things this way (with $x$ discretized) should help track down what's really going on with the $\delta(x-x')$ factor.
A: Consider the more general case of a (not necessarily local) field redefintion of the form:
\begin{equation}
\phi(x)=\int d y \,\,f(x,y)\,g\big(\theta(y)\big)\tag{1}
\end{equation}
For example, in the question above we have
\begin{equation}
\phi(x)=\int d y \,\,\delta(x-y)\,\sinh\big(\theta(y)\big)
\end{equation}
which is an example of a local field redefinition. 
In the discrete case where we think of the path integral as taking place on a lattice, Eq.(1) takes the form, where $\phi_i$ is shorthand for the discrete variable $\phi(x_i)$:
\begin{equation}
\phi_i=\sum_j F_{ij}\, g(\theta_j) \tag{2}
\end{equation}
where $F_{ij}=F(x_i,x_j)$ can be thought of as a matrix and $F_{ij}=\delta_{ij}$ corresponds to the case of a local transformation. Discretizing the path integral we can write the change of variables in Eq.(2) as (here $\wedge$ denotes the wedge product, which is always present for the tensor density $d^dx$ but I make it explicit here only to make the presence of the Jacobian apparent)
\begin{align}
\int \mathcal{D}\phi&=\int d\phi_1\wedge d\phi_2\wedge...\wedge d\phi_n\\
&= \int \frac{1}{n!}\epsilon_{i_1...i_n}d\phi^{i_1}\wedge...\wedge d\phi^{i_n}\\
&=\int \frac{1}{n!}\epsilon_{i_1...i_n}\,\,\frac{\partial\phi^{i_1}}{\partial\theta^{i_1'}}...\frac{\partial\phi^{i_1}}{\partial\theta^{i_n'}}d\theta^{i_1'}\wedge...\wedge \,d\theta^{i_n'}\\
&=\int \frac{1}{n!}\epsilon_{i_1...i_n}\,\,\bigg(F^{i_1i_1'}\frac{d g(\xi)}{d\xi}\bigg\vert_{\xi=\theta_{i_1'}}\bigg)...\bigg(F^{i_ni_n'}\frac{d g(\xi)}{d\xi}\bigg\vert_{\xi=\theta_{i_n'}}\bigg)d\theta^{i_1'}\wedge...\wedge \,d\theta^{i_n'}\\
&=\int\frac{1}{n!}\epsilon_{i_1'...i_n'}\text{Det}\bigg[F^{ii'}\frac{d g(\xi)}{d\xi}\bigg\vert_{\xi=\theta_{i'}}\bigg]d\theta^{i_1'}\wedge...\wedge \,d\theta^{i_n'}\\
&=\int\text{Det}\bigg[F^{ii'}\frac{d g(\xi)}{d\xi}\bigg\vert_{\xi=\theta_{i'}}\bigg]d\theta^{i_1'}\wedge...\wedge \,d\theta^{i_n'}\\
&=\int d\theta^{i_1'}\wedge...\wedge \,d\theta^{i_n'}\,\,\exp\bigg\lbrace\text{Tr}\bigg(\log\bigg[F^{ii'}\frac{d g(\xi)}{d\xi}\bigg\vert_{\xi=\theta_{i'}}\bigg]\bigg)\bigg\rbrace\tag{3}
\end{align}
So for example if we have a local field redefintion $F_{ij}=\delta_{ij}$ then we encounter the term $Tr(\log(\delta_{ij}))=\log(n)$, where $n$ is the number of lattice sites. In the continuous case where $F_{ij}\rightarrow f(x,y)=\delta^{d}(x-y)$ we encounter the highly singular term 
\begin{equation}
Tr(\log(\delta^d(x-y)))=\int d^dx\,\,\log\bigg(\delta^{d}(x-x)\bigg)=\int d^dx\,\,\log\bigg(\delta^{d}(0)\bigg)
\end{equation}
So to answer the original question, I think the measure transforms as:
\begin{align}
\mathcal{D}\phi=\mathcal{D}\theta\,\,\exp\bigg(\int dx\,\log\big(\delta(0)\big)+\log\big(1+3c\theta^2)\big)\bigg)\tag{1}\\
\mathcal{D}\phi=\mathcal{D}\theta\,\,\exp\bigg(\int dx\,\log\big(\delta(0)\big)+\log\big(3c\theta^2\big)\bigg)\tag{2}\\
\mathcal{D}\phi=\mathcal{D}\theta\,\,\exp\bigg(\int dx\,\log\big(\delta(0)\big)+\log\big(\cosh(\theta(x))\big)\bigg)\tag{3}
\end{align}
Apparently one can ignore the $\delta^d(0)$ when working in dimensional regularization as we can interpret $\delta^d(0)$ as the volume of spacetime, which in $d-\epsilon$ dimensions is
\begin{equation}
\delta^{d}(0)=\frac{2\pi^{d/2}}{\Gamma(d/2)}\frac{\Gamma(-d)}{\Gamma(1-d)}
\end{equation}
which is $\frac{1}{\epsilon}$ dependent and hence we can use $\frac{1}{\epsilon}$ dependent counterterms to get rid of this term. However $\exp(\log(g'))$ terms still remain and it is unclear to me how to show that correlation functions will be unaffected by this term.
