  The formulation of a new analysis on a zero measure Cantor set $C (\subset
I=[0,1])$ is presented. A non-archimedean absolute value is introduced in $C$
exploiting the concept of {\em relative} infinitesimals and a scale invariant
ultrametric valuation of the form $\log_{\varepsilon^{-1}} (\varepsilon/x) $
for a given scale $\varepsilon>0$ and infinitesimals $0<x<\varepsilon, x\in
I\backslash C$. Using this new absolute value, a valued (metric) measure is
defined on $C $ and is shown to be equal to the finite Hausdorff measure of the
set, if it exists. The formulation of a scale invariant real analysis is also
outlined, when the singleton $\{0\}$ of the real line $R$ is replaced by a zero
measure Cantor set. The Cantor function is realised as a locally constant
function in this setting. The ordinary derivative $dx/dt$ in $R$ is replaced by
the scale invariant logarithmic derivative $d\log x/d\log t$ on the set of
valued infinitesimals. As a result, the ordinary real valued functions are
expected to enjoy some novel asymptotic properties, which might have important
applications in number theory and in other areas of mathematics.
