Does $E(X|\min(X,k))$ converges to $X$ in $L^1$ or almost surely or neither?

Question

$P(X=k)=1/(2^k)$, $k=1,2,3... ~~~ Y_k=E(X|\min(X,k))$

Does $Y_k$ converges to $X$ in $L^1$ or almost surely or neither? Can somebody give a proof?

Answer 1

For every integrable random variable $X$ and every real number $x$ such that $P[X\geqslant x]\ne0$, $$ E[X\mid\min(X,x)]=X\,\mathbf 1_{X\lt x}+E[X\mid X\geqslant x]\,\mathbf 1_{X\geqslant x}. $$