When it comes to household income, progressives tend to start with their intuitive understanding of fairness (i.e., some people have a lot more income than others), move to the solution (redistribution of income and wealth from those who have more to those who have less), and only then to develop a metric that justifies implementing their solution: income inequality.
Because of this roundabout approach, you rarely hear progressives argue that income inequality is a problem since for them it just is an injustice — and that wealth redistribution is the primary solution. When conservatives and libertarians disagree about whether it even is an issue we should be concerned with, we are considered heart-hearted apologists for an immoral capitalist system.
The truth, however, is that we don’t care about income inequality because relative differences in income tell us nothing about fairness or the just distribution of wealth. What we care about — what everyone should care about — is whether people have adequate opportunities to increase their household’s income, and hence, improve their standard of living. While there is no truly adequate gauge to measure such opportunities, we can get a fair estimate based on measurements of social mobility.