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Im not entirely sure what this definition means, whilst I'm reading up.

Let $X_n \in$ some sigma algebra $\mathcal{F}$.

$X_n \uparrow X = X_n \subseteq X_{n+1}, \forall n \in \mathbb{N}$ and $\cup X_n = X.$

All that was written in my notes was the above line (which seems to be conditions?) But does $X_n \uparrow X$ mean that $X_n$ converges to $X$?

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    Thanks guys, great help, I understand it now!2012-03-12

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The line you quote is a definition of $X_n \uparrow X$ although I agree that this is not very clear. That is, the line states that $X_n \uparrow X$ means $X_n \subseteq X_{n+1}$ and $\bigcup X_n = X$.

You should think of $\uparrow$ as meaning increasing pointwise convergence. More generally, if $f_n$ is a sequence of measurable real-valued functions, we say that $f_n \uparrow f$ if $f_n \le f_{n+1}$ pointwise and $f_n \to f$ pointwise. This reduces to the above definition if we take $f_n = 1_{X_n}, f = 1_X$.

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    And writing an equal sign for "means" just causes you confusion a few days later when you try to read your notes!2012-03-12