Dash 8 Captain reported a fuel imbalance was indicated after takeoff. The crew elected to continue to the destination; believing the imbalance was due to an indicating anomaly; not an actual fuel imbalance. Post-flight; it was determined there was a large fuel imbalance which had been developing during the flight.

Narrative

This report is to document an 800 lb fuel imbalance we experienced immediately after take off from ZZZ to ZZZ1. Aircraft X was fueled prior to departure to an indicated 5;900 lbs. Holding short of the runway prior to departure approximately 50 lbs of fuel was transferred from the right to the left tank to achieve 0 lb imbalance indication. This transfer took about :30 seconds to accomplish. Immediately after a normal take off I noticed the fuel quantity numbers changing rapidly. I mentioned to the First Officer we don't have it yet but we are going to get a fuel imbalance message on the ED. At acceleration height we observed a flashing balance message in amber on the ED. I assumed we had a fuel quantity indication problem and elected not to transfer any fuel believing it was balanced perfectly minutes prior. My First Officer wondered if we had developed a massive fuel leak. Observing the quantity indicators it appeared we did not as the two quantity's were decreasing together in lock step; at a normal rate and what appeared to be exactly 400 more on the left and 400 less on the right. We contacted the flight attendants and requested they observe the right wing for evidence of a fuel leak. Referencing the checklist under transfer problems ended with the statement to land at solid bug if imbalance exceeded 600 lbs. First Officer was hand flying the aircraft and trims were centered and no abnormal control forces were noted. I contacted Dispatch to inform them we had a suspected fuel quantity discrepancy. Informed Maintenance Control we had a suspected fuel quantity discrepancy with a 800 lb imbalance left wing heavy. Plan was to proceed to ZZZ1. First Officer hand flew the aircraft from enroute phase to a normal landing somewhere between open bug and solid bug. The First Officer and I believed we had an indication anomaly. That we had perfectly balanced indication prior to departure. That the variance was exactly +400 on the left and -400 on the right. That the trims were centered. That the plane felt normal in control forces.I was humbled and embarrassed when the mechanic lowered the left wing mag stick with my penny and it indicated 360 gallons; approximately 2400 lbs; the same as what was indicated on the left quantity indicator. The walk around to the right wing was marked with much emotion with the realization I/we had disregarded correct information potentially compromising the safety of flight. The right wing mag stick revealed 170 gallons; approximately 1140 lbs resulting in a more than 1200 lb imbalance. The right fuel quantity gauge was indicating approximately 1650 lbs at that time so there was an indication problem. A problem that existed on the ground in ZZZ when the plane was refueled. A problem that remained during the taxi; in flight and remained on the ground in ZZZ1. The right wing quantity I do not believe ever read correctly. We had just started our duty day in ZZZ.Not to discount the possibility you are getting correct information and it was incorrect before. Confirmation bias is strong.