Young infants appear to have a gap in their protection against measles, from around two to three months old until they are vaccinated at 12 months of age...This is because the level of antibodies infants get from their mother drops over time, leaving them susceptible until they are vaccinated.So parents who choose not to vaccinate their kids risk infecting other people's infants.
The study was looking at maternal antibodies and had some other interesting results. Women who had been vaccinated themselves had fewer antibodies than those who got their immunity from having the disease, and so did their kids, and then mom's antibodies fade away in the first few months, much faster in kids of vaccinated moms. "The researchers found no significant impact of breastfeeding, birth weight, educational level, caesarean section or day care attendance on the duration of maternal antibodies" in the kid.
So immunity from measles is stronger in kids of women who got the immunity from the disease. The question is whether the immunity acquired through vaccination is enough to prevent the disease, and it seems to be.