Entropy of the Mixture of Sources and Entropy Dimension

Suppose that we are given two sources
$S_{1}$
,
$S_{2}$
which both send us information from the data space
$X$
. We assume that we lossy-code information coming from
$S_{1}$
and
$S_{2}$
with the same maximal error but with different alphabets
${\cal P}_{1}$
and
${\cal P}_{2}$
, respectively. Consider a new source
$S$
which sends a signal produced by source
$S_{1}$
with probability
$a_{1}$
and by source
$S_{2}$
with probability
$a_{2}=1-a_{1}$
. We provide a simple greedy algorithm which constructs a coding alphabet
${\cal P}$
which encodes data from
$S$
with the same value of maximal error as single sources, such that the entropy
${h}(S;{\cal P})$
satisfies:
$$ {h}(S;{\cal P})\leq a_{1}{h}(S_{1};{\cal P}_{1})+a_{2}{h}(S_{2};{\cal P}_{2})+1. $$
In the proof of the aforementioned formula, the basic role is played by a new equivalent definition of entropy based on measures instead of partitions. As a consequence, we decompose the entropy dimension of the mixture of sources in terms of the convex combination of the entropy dimensions of the single sources. In the case of probability measures in
$\BBR^{N}$
, this allows us to link the upper local dimension at point with the upper entropy dimension of a measure by an improved version of Young estimation.