Connection between Haars measure Hilbertspace and Quantum mechanics.

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Iam a physics student,I don't have any background in measuring theory. What I know is group theory and some priliminary basics of topology. In a book by MV Altaisky I seescreen shot from the book of wave lets by MV Altaisky.

I don't understand how he get expression 2.1and 2.2 and also I don't get how he connect it to famous quantum mechanical formulae 2.3. Can any one please spend some of your valuable time to help me. Thanks in advance.