Grand and small norms in Lebesgue spaces

We present a new approach to the definition of grand and small Lebesgue spaces. This in particular allows to include into consideration the extreme exponents p=1$$ p equal to 1 $$ and p=& INFIN;$$ p equal to \infty $$. Basically, the study of the extreme exponent case is the main result of the article. However, we expect that our general construction for the norms in grand and small Lebesgue spaces will imply further studies of the spaces and operators in these spaces in such a general setting.