Let $\mu$ be the Lebesgue measure on the segment $[0,1]$. Let $(f_n)$ be a sequence in $L^{2}([0,1])$ such that $f_n\rightarrow f$ p.w. and $f_{n+1}\geq f_{n}$. Does $f_n\rightarrow f$ also in the 2-norm? (what if $(f_n)$ is a sequence of simple functions?)
Does pointwise monotone convergence imply 2-norm convergence?
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real-analysis
measure-theory
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1I'm surprised that you know what $L^2$ is without knowing the monotone convergence theorem. Where did you learn this material from? By the way, what does "p.w." mean? – 2011-08-21
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0p.w. = pointwise I guess – 2011-08-21
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2But the pointwise limit of $L^2$ functions need not be $L^2.$ Consider the family given by $f_n(x) = \frac{1}{\sqrt{x+1/n}}.$ Then $f_n$ is a positive, nondecreasing sequence of $L^2([0,1])$ functions, but the pointwise limit ${1}/{\sqrt{x}}$ is not $L^2.$ Moreover, neither is $|f_n -f|$ for any $n.$ The monotone convergence theorem only answers the $L^1$ analog of mathfreaks question. – 2011-08-21