Convergence in the Radon metric implies weak convergence of measures:

It is therefore plain that uniform convergence implies pointwise convergence.

Hence, convergence in mean square implies convergence in mean.

Usually, convergence in distribution does not imply convergence almost surely.

These two conditions imply convergence.

Therefore convergence in norm implies weak convergence while the converse is not true in general.

Global convergence in measure implies local convergence in measure.

In particular, for series with values in any Banach space, absolute convergence implies convergence.

When G is complete, absolute convergence implies unconditional convergence.

Cohesion also has a fundamental role to play in progress towards economic and monetary union, which implies convergence between the economies of the Member States.