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What happens if we relax the assumption \(a_1,…,a_n\geq 0\) in ???
Let \(f,f_1,f_2,…:ℝ^d→[0,∞]\) such that \(f_n→f\) \(\lmg \)-almost everywhere. Does \(∫f_n\intd \lmg \) converge?
With \(f,f_n\) from ??, if \(∫f_n\intd \lmg \) converges, what are the possible values for its limit in terms of \(∫f\intd \lmg \)?
With \(f,f_n\) from ??, under which conditions does \(∫f_n\intd \lmg \) necessarily converge to \(∫f\intd \lmg \)?
Is ?? also true for \(∞\) instead of \(n\)?