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$F$ is the conditional expectation function (the "continuation value") and our approximate of this using $M$ basis functions is $F_M$... but in the paper, they have this theorem:

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What is $F_X$? It has not been defined in the paper.

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From the arguments in the lines following the proposition in this paper where the proposition is made; it really looks as if $F_X$ is a typo which should actually read $F_M$, which stands for an approximation of $F$ using the first $M$ basis functions. It is argued that:

The key to this result is that the convergence of $F_M(w, t)$ to $F(w;t)$ is uniform on (0,m) when the indicated integrability conditions are met.

which clearly points to that the integrabilitiy condition in the proposition made for $F$ and $F_X$ really was meant for $F$ and $F_M$

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