I carry a plank of wood along to the carpenter and ask her/him to cut me another to the same length. Next time I'm a bit more savvy and mark off a length of string against the plank, roll it up and put it my pocket, and take that along to the carpenter instead. Third time I'm even cuter, I measure the length in numbers on a tape measure, and just go along with nothing but tiny puffs of air on my lips or marks on a bit of paper.
Each step progressively liberates information from its cumbersome physical representation or embodiment for the purposes of transmission, storage and eventual reconstitution. Living things do the same with DNA.
So surely one kind of measurement we're concerned with must be a ratio demonstrating the degree to which this liberation is accomplished, such as gigabytes per chip or per second in electronics, or quantity per number in arithmetic. For example that ratio is 37 for the number 111 since we have just three symbols expressing the quantity one hundred and eleven which in primitive times would have had to be transmitted as a hundred and eleven symbols.
This approach also means that information can be instructions as well as descriptions, those two words simply represent different directions: to and from physical embodiment.