@RalphLambrecht here's you could wire it up:
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1: Connect battery + terminal (the smaller one; indicated on the battery with a "+") to the positive terminal of the light meter's battery connector. This is the crown-based connector.
2: Connect the light meter's negative terminal to the positive probe (red one) of the multimeter.
3: Connect the multimeter's black measurement probe to the negative terminal of the battery.
In the diagram above I've indicated with arrows how the current flows. It goes from the battery, through the light meter, through the multimeter and then back into the battery.
In reality, it looks like this:
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* The multimeter needs to be set to current measurement mode. Otherwise the circuit doesn't work. On some multimeters, this may involve plugging the positive probe (red one) into a different position than when measuring voltages. However, in most multimeters, this is not the case for small currents as we measure here. Refer to the manual of your multimeter.
* Depending on the design of your multimeter, it may have to be switched on for the circuit to work. I.e. with some multimeters, if the multimeter is turned off, no current will flow and the light meter cannot be turned on.
* Try measuring the current first at a relatively high measurement range, for instance a 200mA range. Depending on what you observe, select a lower range for better resolution.
Never attempt to measure a current larger than the range the multimeter is set to as it may burn out an internal safety fuse. For the same reason,
never try to measure a short-circuit current.
In the photo above, I've set the metering range to the lowest this particular cheap multimeter has to offer (200uA/microamperes). As the meter turns on, there's a brief peak, but the idle current with the display on is around 80uA as can be seen here. This looks like a perfectly normal/healthy current. The power-down current of the light meter will be less and may even be too low to register on the multimeter, unless it's a very sensitive one. As long as you measure a current lower than e.g. 200uA, the light meter itself is not the cause of your battery run-down problem. You can calculate how long a typical battery will last if you know the current that's being drawn.