Pasteurization and Bioavailability of Antioxidants in Beet Juice
So, when they subjected "ready to drink" beetroot juice to thermal pasteurization, they found the betacyanin and betaxanthin (our major antioxidants, and pigment molecules) content to be 39.9 and 42.28% degraded, respectively (1). Their conclusion was as follows:


  Standardization of process condition and
  quality degradation of beetroot juice due to thermal
  pasteurization was studied. The standardized T2
   batch
  with total heating time (fh) of 720 s thermal in-pack
  pasteurization leads to the minimal degradation of
  color, betalain content (Betacyanin and Betaxanthin),
  antioxidant activity and complete inactivation of
  micro flora of beetroot juice. The color, betalain
  content (Betacyanin and Betaxanthin) antioxidant
  activity and sensory of the beetroot juice was
  significantly (p &lt; 0.05) reduced during 180 days
  ambient (27-30°C) storage, but still the quality of the
  juice was adequate upto 180 days. We concluded that
  thermal pasteurization of 96 °C for a total heating time
  (fh) of 720 seconds with P-values of 11.16 would be
  a good method to produce microbiologically stable
  beetroot juice with the retention of quality attributes.


So our goal is to get a juice that retains as many "good" attributes as possible, while producing a product that isn't going to get anyone a foodborne illness. As for the net effect of pasteurization on fruit juice in general, im still coming to a conclusion there. The process shouldn't render too much of the fruit/vegetable ineffective, though I'd still expect some degradation either in storage or processing.