If $f_n$ converges to $f$ pointwise, then the Riemann integral of $f_n$ over a bounded interval $[a,b]$ need not converge to the Riemann integral of $f$ over $[a,b]$. But if $f_n$ converges to $f$ uniformly, then the Riemann integral of $f_n$ over $[a,b]$ does converge to the Riemann integral of $f$ over $[a,b]$.
However, even if $f_n$ converges to $f$ uniformly, the improper Riemann integral of $f_n$ over an unbounded interval like $[a,\infty]$ need not converge to the improper Riemann integral of $f$ over that unbounded interval. So my question is, does there exist some notion of convergence of functions stronger than uniform convergence such that if $f_n$ converges to $f$ under that notion, then the improper Riemann integral of $f_n$ over any unbounded interval converges to the improper Riemann integral of $f$ over that unbounded interval?