  In this letter, we explain how the U(1) BF measure can be related to the
Fourier transform of a Dirac distribution defined on the $\mathbb{Z}$-module of
quantum fields. Then, we revisit the U(1) BF partition function with the help
of this Dirac distribution and finally shed light on a natural relation between
the U(1) BF and Chern-Simons theories.
