  We construct a Banach rearrangement invariant norm on the measurable space
for which the finiteness of this norm for measurable function (random variable)
is equivalent to suitable tail (heavy tail and light tail) behavior.
  We investigate also a conjugate to offered spaces and obtain some embedding
theorems.
  Possible applications: Functional Analysis (for instance, interpolation of
operators), Integral Equations, Probability Theory and Statistics (tail
estimations for random variables).
