Regarding spinel-type lithium manganese-based composite oxide (LMO) to be used as a positive electrode active substance material for lithium battery, a novel LMO is provided, which is capable of maintaining discharge capacity even if charging and discharging are repeated under high temperatures. An LMO in which the crystallite size is 250 nm to 350 nm, the strain is 0.085 or less and the specific surface area increase rate when placed in water at 25° and pH 7 and ultrasonically dispersed at 40 W ultrasonic intensity for 600 seconds is 10.0% or less, can prevent a decrease in the output that accompanies the repetition of charging and discharging while at a high temperature.