I am trying to show that the stationary distribution for a Markov Chain on a continuous state space can be obtained by building a transition density kernel, which obeys the detailed balance rule where $P_{t+1|t}(X_{t+1}=i|X_{t}=j)P_{t}(X_{t}=j) = P_{t+1|t}(X_{t+1}=j|X_{t}=i)P_{t}(X_{t}=i)$. The Markov Chain is time-homogeneous so $P_{t+1|t}(X_{t+1}=i|X_{t}=j)=T(X_{t+1}=i|X_{t}=j)$ is independent of time step $t$ and $T$ is the continuous transition kernel (density kernel). I suppose that the distribution at the time $t$ is the stationary distribution $\pi$, which is $P_{t}(X_{t}=i) = \pi(i)$ and I aim to show that $P_{t+1}(X_{t+1}=i) = \pi(i)$ given the detailed balance.
I am currently proving this already but I am not sure whether my way of proof is mathematically valid.
Here is what I am doing:
1) I calculate $P_{t+1}(X_{t+1}=i)$ as $P_{t+1}(X_{t+1}=i) = \int_{-\infty}^{\infty} P_{t+1|t}(X_{t+1}=i|X_{t})P_{t}(X_{t})dX_{t} = \int_{-\infty}^{\infty} P_{t+1,t}(X_{t+1}=i,X_{t})dX_{t}$
2) According to the detailed balance, for each $i,j \in \mathbb{R}$, it is $P_{t+1|t}(X_{t+1}=i|X_{t}=j)P_{t}(X_{t}=j) = P_{t+1|t}(X_{t+1}=j|X_{t}=i)P_{t}(X_{t}=i)$. This means that the joint distribution function is symmetrical: $P_{t+1,t}(X_{t+1}=i,X_{t}=j) = P_{t+1,t}(X_{t+1}=j,X_{t}=i)$. Here, I think about the joint distribution as an uncountable sized, symmetric matrix. Therefore the integration over the row "belonging" to $X_{t+1}=i$ is equal to the integration over the column "belonging" to $X_{t}=i$. According to this logic it must be $P_{t+1}(X_{t+1}=i) = \int_{-\infty}^{\infty} P_{t+1,t}(X_{t+1}=i,X_{t})dX_{t} = \int_{-\infty}^{\infty} P_{t+1,t}(X_{t+1},X_{t}=i)dX_{t+1} = P_{t}(X_{t}=i)$.
3)Since $P_{t}(X_{t}=i) = \pi(i)$ we have that $P_{t+1}(X_{t+1}=i) = \pi(i)$ and this ends the proof.
The part 2 is where I am feeling uneasy about my way of proof. I know that the symmetry induced on the joint distribution of consecutive random variables must yield the change of integration variables in the step 2 as valid. This is indeed the case, if we had a finite state space Markov Chain. The joint distribution would be a finite sized matrix and the symmetry would immediately tell that the $i$th row and $i$th column are equal. But I am not comfortable with this way of thinking on a continuous state space, mainly because of my shallow mathematical background.
What would be a more mathematically rigorous way to show that the step 2 is correct? How should we show that integrals in the step 2 are equal to each other?
Thanks in advance.