On Peskin and Schroeder's QFT book, page 284, the book derived two-point correlation functions in terms of function integrals. $$\left\langle\Omega\left|T \phi_H\left(x_1\right) \phi_H\left(x_2\right)\right| \Omega\right\rangle=$$ $$\lim _{T \rightarrow \infty(1-i \epsilon)} \frac{\int \mathcal{D} \phi \phi\left(x_1\right) \phi\left(x_2\right) \exp \left[i \int_{-T}^T d^4 x \mathcal{L}\right]}{\int \mathcal{D} \phi \exp \left[i \int_{-T}^T d^4 x \mathcal{L}\right]} ,\tag{9.18} $$
Then, in the following several pages, the book calculate the numerator and denominator separately, using discrete Fourier series, and then take the continue limit.
Finally, the book obtain the calculation result of eq.(9.18) in eq.(9.27): $$ \left\langle 0\left|T \phi\left(x_1\right) \phi\left(x_2\right)\right| 0\right\rangle=$$ $$\int \frac{d^4 k}{(2 \pi)^4} \frac{i e^{-i k \cdot\left(x_1-x_2\right)}}{k^2-m^2+i \epsilon}=D_F\left(x_1-x_2\right). \tag{9.27}$$ I am troubled for the L.H.S of (9.27), why $\left\langle\Omega\left|T \phi_H\left(x_1\right) \phi_H\left(x_2\right)\right| \Omega\right\rangle$ became $\left\langle 0\left|T \phi\left(x_1\right) \phi\left(x_2\right)\right| 0\right\rangle$?
Also, the $\phi$ in eq.(9.27) maybe in interaction picture, which is different with Heisenberg picture $\phi_H$ in general case.
A parallel analysis is also in Peskin and Schroeder's QFT book, on page 87, in eq.(4.31) $$ \langle\Omega|T\{\phi(x) \phi(y)\}| \Omega\rangle=\lim _{T \rightarrow \infty(1-i \epsilon)} \frac{\left\langle 0\left|T\left\{\phi_I(x) \phi_I(y) \exp \left[-i \int_{-T}^T d t H_I(t)\right]\right\}\right| 0\right\rangle}{\left\langle 0\left|T\left\{\exp \left[-i \int_{-T}^T d t H_I(t)\right]\right\}\right| 0\right\rangle} .\tag{4.31}$$
And also we only considered the numerator of (4.31) in later analysis, why?