Assume I have a random process $x_1,x_2,\ldots,x_N$ defined on a probability space $(\Omega,G,P)$.
Assume I have the expectation $$E_{x_1}\large[x_1+E_{x_2\vert x_1}\large[x_2+E_{x_3\vert x_1,x_2}\large[x_3+\cdots\large]\large]\large]$$ where $E_{x_t\vert x_{t-1},\ldots,x_{1}}$ I mean the expectation of $x_t$ conditional on the past realizations.
Can this be written as a sum of expectetations rather than nested expectations? For example, is the following correct? $$E_{x_1}\large[x_1\large]+E_{x_2}\large[x_2\large]+E_{x_3}\large[x_3\large]+\cdots$$
My assumption is that this is possible if we do not write conditional expectations, and that it follows from the tower property $$E[x_t]=E_{x_{t-1}}[E_{x_t\vert x_{t-1}}[X_t]]$$
Edit: Measure theoretical version: Assume a sequence of sub-sigma algebras $G_1\subset G_2 \subset \cdots G$ and that the random process is adapted to this sequence.
$$E\large[x_1+E\large[x_2+E\large[x_3+\cdots\vert G_3\large]\vert G_2\large]\vert G_1\large]$$
I suppose the tower property allows me to write $$E[x_t]=E\large[E\large[x_t\vert G_t\large]\vert G_{t-1}\large]$$ and then the above follows.
