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I am having some trouble getting to grips with the intuition behind Quadratic Variation with regards to Brownian Motion and $L^2(P)$ convergence in the construction of the Ito integral. My questions are as follows

$1)$ I don't understand what the quadratic variation of a Brownian motion tells us? Why is it important? Why do we look at it at all?

$2)$ The definition of the Ito integral is given as

$$ \displaystyle{\int^{T}_{S} f(t,\omega)dB_t{(\omega)}=\lim_{n \to \infty}\int^{T}_{S}\phi_n(t,\omega)dB_t(\omega)} \ \ \ (Limit \ in \ L^{2}(P))$$

Where $\{\phi_n\}$ is a sequence of elementary functions such that $$ (a) \ \ E\bigg[\int^{T}_{S}(f(t,\omega) - \phi_n(t,\omega))^2dt\bigg] \ \to \ 0 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ as \ \ n \ \to \ \infty$$ Why do we specifically require $L^2(P)$ convergence? what is so important about $L^2(P)$ as opposed to all other $L^p(P)$ spaces?

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