About positive part of difference fuctions. What can we Say that $(f-g)^{+}$?  $(f-g)^{+}=(f^{+}+(-g)^{+})=f^{+}-g^{-}$?

The positive and the negative part of a function are non negative function. In general it is not true that $ (-g)^+ = - g^-$. It is true if and only if $g=0$.
But we know (see for instance https://en.wikipedia.org/wiki/Positive_and_negative_parts) that for every function 
$$ f^+ = \frac{ | f | +f }{2} \quad f^- = \frac{ | f | -f }{2}.$$
Thus 
$$(f-g)^+ = \frac{|f-g| + f-g }{2} \leq \frac{|f| + |g| + f-g }{2} = \frac{|f| + f}{2} + \frac{|g| -g }{2} = f^+ + (-g)^+.$$
