On a finite-measure set, pointwise a.e. convergence is almost uniform convergence: for every δ > 0 there is a subset F of E with μ(E\F) < δ on which f_n → f uniformly. Fails on σ-finite spaces without the finiteness hypothesis (take f_n =…
On a finite-measure set, pointwise a.e. convergence is almost uniform convergence: for every δ > 0 there is a subset F of E with μ(E\F) < δ on which f_n → f uniformly. Fails on σ-finite spaces without the finiteness hypothesis (take f_n =…