  Although there doesn't exist the Lebesgue measure in the ball $M$ of $C[0,1]$
with $p-$norm, the average values (expectation) $EY$ and variance $DY$ of some
functionals $Y$ on $M$ can still be defined through the procedure of limitation
from finite dimension to infinite dimension. In particular, the probability
densities of coordinates of points in the ball $M$ exist and are derived out
even though the density of points in $M$ doesn't exist. These densities include
high order normal distribution, high order exponent distribution. This also can
be considered as the geometrical origins of these probability distributions.
Further, the exact values (which is represented in terms of finite dimensional
integral) of a kind of infinite-dimensional functional integrals are obtained,
and specially the variance $DY$ is proven to be zero, and then the nonlinear
exchange formulas of average values of functionals are also given. Instead of
measure, the variance is used to measure the deviation of functional from its
average value. $DY=0$ means that a functional takes its average on a ball with
probability 1 by using the language of probability theory, and this is just the
concentration without measure. In addition, we prove that the average value
depends on the discretization.
