I am following section 9.2 in Peskin and Schroeder in which the Feynman rules are derived for scalar fields.
They define (in eqn (9.14), page 282) the transition amplitude from $\vert\phi_a\rangle$ to $\vert\phi_b\rangle$ in time $T$ to be
$$\langle\phi_b\vert e^{-iHT}\vert\phi_a\rangle = \int \mathcal{D}\phi \exp\left[i\int_0^Td^4x\mathcal{L}\right]\tag{1}.$$
Then (on page 289) they deal with the $\phi^4$ theory and use the expansion
$$\exp\left[i\int_0^Td^4x\mathcal{L}\right] = \exp\left[i\int_0^Td^4x\mathcal{L_0}\right]\left(1-i\int d^4x \frac{\phi}{4!}\phi^4+\cdots\right)\tag{2}$$
Each term on the RHS is indeed of the form
$$\int \mathcal{D}\phi\; \phi(x_1)\cdots\phi(x_n) \exp\left[i\int_0^Td^4x\mathcal{L_0}\right],$$
which appears in an easy generalization of the formula (eqn (9.18), page 284)
$$\langle\Omega\vert T\phi_H(x_1)\phi_H(x_2)\vert\Omega\rangle = \lim_{T\to\infty(1-i\epsilon)} \left(\frac{\int \mathcal{D}\phi\; \phi(x_1)\phi(x_2) \exp\left[i\int_{-T}^Td^4x\mathcal{L_0}\right]}{\int \mathcal{D}\phi\ \exp\left[i\int_{-T}^Td^4x\mathcal{L_0}\right]}\right)\tag{3}$$
To derive the Feynman rules, we want to swap each term in (2) for a correlation function.
What happens to the denominator of (3)?
Also, should I worry about taking $T\to\infty(1-i\epsilon)$ instead of $T\to \infty$ (which would the natural thing to do in (1))?