I Pointwise convergence in Lp space

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I'm reading about ##L^p## space and there's a remark after a proposition that confuses me.
Proposition 4.6 Let ##p\in[1,\infty)## and let ##(f_n)## be a convergent sequence in ##L^p(E,\mathcal A,\mu)## with limit ##f##. Then there is a subsequence ##(f_{k_n})## that converges pointwise to ##f## except on a measurable set of zero ##\mu##-measure.

Remark The result also holds for ##p=\infty##, but in that case there is no need to extract a subseqeuence, as the convergence in ##L^\infty## is equivalent to uniform convergence except on a set of zero measure.

Let us mention a useful by-product of Proposition 4.6. If ##(f_n)## is a convergent sequence in ##L^p(E,\mathcal A,\mu)## with limit ##f##, and if we also know that ##f_n(x)\to g(x)## ##\mu(\mathrm{d}x)##-a.e., then ##f=g## ##\mu## a.e.

I fail to see why the "by-product" of Proposition 4.6 is true. Isn't $$f_n\stackrel{L^p}{\to} f \text{ and } f_n(x)\to g(x) \ \mu(\mathrm{d}x)\text{-a.e.}$$ the same statement? I am confused about how to apply Proposition 4.6 to prove the "by-product" statement. Any help is greatly appreciated.
 
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Ok, I'm a bit tired. Here's what I've gathered:

If ##(f_n)## converges to ##f## in ##L^p##, then by Proposition 4.6 there is a subsequence ##f_{n_k}(x)\to f(x)## pointwise for ##\mu##-almost every ##x##. Since we also have that ##f_n(x)\to g(x)## pointwise for ##\mu##-almost every ##x##, and every subsequence of a (almost everywhere) convergent sequence converges to the same limit, it's clear that ##f=g## ##\mu##-a.e.
 
A sphere as topological manifold can be defined by gluing together the boundary of two disk. Basically one starts assigning each disk the subspace topology from ##\mathbb R^2## and then taking the quotient topology obtained by gluing their boundaries. Starting from the above definition of 2-sphere as topological manifold, shows that it is homeomorphic to the "embedded" sphere understood as subset of ##\mathbb R^3## in the subspace topology.

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