This paper investigates stochastic convergence in income inequality across Australian states and territories since the end of World War II by utilising the LM and RALS-LM unit root tests that allow for endogenously determined structural breaks. We find that income inequality for Australia's capital city---the Australain Capital Territoy---converges to a stable steady-state when we account for endogenously determined trend-breaks. The null hypothesis of a unit root is rejected when utilising the two-break unit root test for New South Wales, Northern Territory, Queensland, South Australia, Tasmania and Western Australia indicating that income inequalities in these states converge to a stable steady-state when accounting for multiple structural breaks. In contrast, the state of Victoria consistently shows evidence of divergence regardless of the number of endogenously determined structural breaks. Structural changes in income inequality across states and territories may be linked to the mining boom, the transition from manufacturing to a service-based economy, changes in government welfare, and favourable changes in tax policies and superannuation.