Flight Crew Air Transport Pilot (ATP)
Confusion; Situational Awareness
Ground Excursion Taxiway
Air Traffic Control Provided Assistance; Flight Crew Became Reoriented
Airport; Human Factors; Weather
Very foggy taxi out prior to sunrise; used appropriate LAX Low Visibility taxi chart.  Because of need to protect the ILS approach area; we were given a non-standard taxi route of Alpha to Golf; hold short of 25L. Then cleared to cross 25L and R on Golf; continue on Bravo to 25R. There are no taxiway centerline lights on Golf; and the fog was so thick that the center taxi stripe was grey washed out. On crossing 25L; the First Officer warned me that I was lined up too far left; which I then noticed; as I had been looking for the taxiway centerline marking; which could not be seen. I did a correction to the right; and asked the First Officer to have LAX Ground have someone come out and check to see that I had not struck any of the taxiway lights.  We waited at #1 for 25R until Ground reported that the area had been checked and there were no lights damaged nor anything out of the ordinary.  Checked the configuration page to ensure no tires indicated any loss of pressure; which they did not and; in consultation with the First Officer; decided to continue the flight. On arrival all tire pressures were still good; and there was no damage noted in a post-parking walkaround of the MLG. Aside from human error; I feel that for visibility this low; it is almost to be expected that poor taxiway centerline markings could lead to being off center. At some airports; the taxiway centerline is marked with a white line; and then a darker colored stripe on each side to set off the white; which will turn gray with age and use; from the gray taxiway surface.  If there is to be no money spent on centerline lights; then perhaps some paint could be used; to help make the centerline more visible in very restricted visibility?
Air Carrier Captain reports a possible taxiway excursion at LAX in dense fog while crossing Runways 25L and R on taxiway Golf. With no centerline lights the centerline stripe is reported as nearly invisible. No taxiway lights are damaged and the flight departs.