Convergence in Measure - Counterexamples

Counterexamples

Let, μ be Lebesgue measure, and f the constant function with value zero.

• The sequence converges to f locally in measure, but does not converge to f globally in measure.

• The sequence where and

(The first five terms of which are ) converges to f locally in measure; but for no x does f_n(x) converge to zero. Hence (f_n) fails to converge to f almost everywhere.

• The sequence converges to f almost everywhere (hence also locally in measure), but not in the p-norm for any .