A measure of compression gain for new symbols in data-compression

In coding theory it is widely known that the optimal encoding for a given alphabet of symbol codes is the Shannon entropy times the number of symbols to be encoded. However, depending on the structure of the message to be encoded it is possible to beat this optimal by including only frequently occurring aggregates of symbols from the base alphabet. We prove that the change in compressed message length by the introduction of a new aggregate symbol can be expressed as the difference of two entropies, dependent only on the probabilities of the characters within the aggregate plus a correction term which involves only the probability and length of the introduced symbol. The expression is independent of the probability of all other symbols in the alphabet. This measure of information gain, for a new symbol, can be applied in data compression methods.