A fine-tuning process for working on a rotational symmetrical or non-rotational symmetrical workpiece surface on a workpiece (W) using at least one fine-tuning operation is carried out in a fine-tuning machine (100) under the control of a control unit (120) of the fine-tuning machine. For the production of material transfer at the planta precision machining tool (115) with geometrically indeterminate cuts is used on the piece surface and a relative movement is created between the fine machining tool (120) and the workpiece (W). A liquid cooling lubricant is introduced into an intervention area between the fine machining tool and the workpiece surface. There will be an online measurement at least ea characteristic of the lubricant introduced relevant to the fine-tuning process to produce a lubricant measurement signal. The refrigerant lubricant measuring signal or a signal derived from it is processed in the control unit (120) of the fine-tuning machine and can be displayed, for example, in the form of a concentration indication on a display unit (130).
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