Before revision:  This space is denoted by L^{p(\cdot) (\Omega, E) }  where the variable exponent p(\cdot) is no longer a given real number like the space L^{p}, but a measurable function .
After revision:  This space is denoted by L^{p(\cdot) (\Omega, E) } where the variable exponent p(\cdot) is no longer a given real number like the space L^{p}, but a random variable, which reflects the possible volatility of the financial markets. The dual representation for this quasiconvex risk measures will also provided .