= Solution
Let $q$ be conjugate to $p$, so $1/q=1-1/p$, including $q=1$ for $p=\infty$. <Hölder's inequality> implies, for every measurable finite-measure <set> $E$,
$$
\int_E|f_n|\,d\lambda\le M\lambda(E)^{1/q},\qquad
\int_E|f|\,d\lambda\le M\lambda(E)^{1/q}.
$$
This proves integrability on $A$ and gives <uniform integrability> there. Choose a <set> $B\subseteq A$ with $\lambda(A\setminus B)\le\delta$ on which convergence is uniform, using <Egorov theorem>. Then
$$
\int_A|f_n-f|\,d\lambda
\le\lambda(A)\sup_B|f_n-f|+2M\delta^{1/q}.
$$
First take $n\to\infty$ and then $\delta\downarrow0$. Thus even the local $L^1$ difference tends to zero, and in particular
$$
\boxed{\int_A f_n\,d\lambda\longrightarrow\int_A f\,d\lambda}.
$$
For $1<p<\infty$, simple <functions> supported on finite-measure <Borel sets> are dense in $L^q(\mathbb R)$. The displayed convergence proves pairing convergence for such <functions>. Approximate any $h\in L^q$ by a simple $h_0$ and use
$$
\left|\int(f_n-f)(h-h_0)\,d\lambda\right|\le2M\|h-h_0\|_q.
$$
By the <duality of Lp spaces>, these are all <continuous linear functionals> on $L^p$, so $f_n$ converges weakly to $f$. It need not converge strongly: $f_n=\mathbf1_{[n,n+1]}$ converges pointwise to zero but has $L^p$ <norm> one. For $p=\infty$ the pairing argument gives weak-star convergence against $L^1$, rather than asserting <weak convergence> against every functional in the much larger full dual.
Back to article page
