There is no such thing as THE inflation rate....the CPI is A measure of inflation. The PPI, the ECI, the PCE, the HICP, the import price index, the export price index, the GDP deflator, the etc are all measures of inflation. As for BLS saying the CPI is not a measure of inflation....from BLS:
Consumer Price Index Frequently Asked Questions
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Various indexes have been devised to measure different aspects of inflation. The CPI measures inflation as experienced by consumers in their day-to-day living expenses; the Producer Price Index (PPI) measures inflation at earlier stages of the production process; the Employment Cost Index (ECI) measures it in the labor market; the BLS International Price Program measures it for imports and exports; and the Gross Domestic Product Deflator (GDP Deflator) measures inflation experienced by both consumers themselves as well as governments and other institutions providing goods and services to consumers. Finally, there are specialized measures, such as measures of interest rates.
The "best" measure of inflation for a given application depends on the intended use of the data. The CPI is generally the best measure for adjusting payments to consumers when the intent is to allow consumers to purchase at today's prices, a market basket of goods and services equivalent to one that they could purchase in an earlier period."
The CPI uses a cost of living approach. It's the most commonly used measure, but even so there are different variations: the CPI-U as the headline number; the CPI-W used for Social Security COLA; the CPI-E, an experimental index based on weights for the elderly; and the CCPI-U, the chained consumer price index.
I'm curious as to how you think measuring the change in prices to maintain a constant standard of living is not a measure of inflation.