how can the purity of water be established?
It depends on what contaminents you are interested in and what the water is to be used for.
One of the most often quoted is total dissolved solids (TDS)
http://en.wikipedia.org/wiki/Total_dissolved_solids
which can be simply found by evaporation of the sample or conductivity of the sample.
Dissolved gases can be found by a number of ways. Gas chromatography of a water sample or of the head space above a sample is used. There are commercial dissolved gas analysers.
Gas chhromography is also used for some organic contaminents.
Many contaminents have their own test or sampling method. There are 'wet chemical' test kits for contaminents such as dissolved oxygen or aluminium.
Hope this helps.