Lebesgue measure

(noun)

The unique complete translation-invariant measure for the $\sigma$-algebra which contains all $k$-cells—in and which assigns a measure to each $k$-cell equal to that $k$-cell's volume (as defined in Euclidean geometry: i.e., the volume of the $k$-cell equals the product of the lengths of its sides).

Examples of Lebesgue measure in the following topics:

  • Continuous Probability Distributions

    • Mathematicians also call such a distribution "absolutely continuous," since its cumulative distribution function is absolutely continuous with respect to the Lebesgue measure $\lambda$.
    • For example, if one measures the width of an oak leaf, the result of 3.5 cm is possible; however, it has probability zero because there are uncountably many other potential values even between 3 cm and 4 cm.

  • Austerity Measures

  • Pickup Notes and Measures

    • If a measure or group of measures has more or fewer beats, the time signature must change.
    • In this case, the first measure may be a full measure that begins with some rests.
    • But often the first measure is simply not a full measure.
    • This shortened first measure is called a pickup measure.
    • If a piece begins with a pickup measure, the final measure of the piece is shortened by the length of the pickup measure.

  • Impact of Measurement Error

    • Measurement error leads to systematic errors in replenishment and inaccurate financial statements.
    • Measurement error is the difference between the true value of a quantity and the value obtained by measurement.
    • In inventory controlling, measurement error is the difference between the actual number of stocks and the value obtained by measurement.
    • Inventory systems can be vulnerable to errors due to overstatements (phantom inventory) when the actual inventory is lower than the measurement or understatements (missing inventory) when the actual stocks are higher than the measurement.
    • In sum, systematic measurement error can lead to errors in replenishment.

  • Units of Measurement for Microbes

    • Measuring microbes presents challenges because they are very small, requiring indirect measures of microbes to understand them better.
    • As a result, measuring them can be very difficult.
    • However, length is not the only measurement that pertains to microbes.
    • DNA is measured in base pairs of DNA.
    • Microbial growth is an important measure in understanding microbes.

  • Collecting and Measuring Data

    • There are four main levels of measurement: nominal, ordinal, interval, and ratio.
    • Nominal measurements have no meaningful rank order among values.
    • Interval measurements have meaningful distances between measurements defined, but the zero value is arbitrary (as in the case with longitude and temperature measurements in Celsius or Fahrenheit).
    • Measurement processes that generate statistical data are also subject to error.
    • Distinguish between the nominal, ordinal, interval and ratio methods of data measurement.

  • Barriers to Managing Control

    • Inaccurate measurement while managing control can happen for a number of different reasons, including:
    • Lack of staff training to determine how to measure the control process
    • All measurement tests need to be tested themselves; an understanding of best practices in taking measurements is critical in project managment.
    • Ensuring timely discovery and reporting of measurements mitigates this risk.
    • Output changes over time, creating room for error in the measurement process.

  • Measuring the Money Supply: M2

    • There is no single "correct" measure of the money supply.
    • Instead there are several measures, classified along a continuum between narrow and broad monetary aggregates.
    • Broader measures add less liquid types of assets (certificates of deposit, etc.).
    • Narrow measures include those more directly affected and controlled by monetary policy, whereas broader measures are less closely related to monetary policy actions.
    • M2 is one of the aggregates by which the Federal Reserve measures the money supply .

  • Scorecard Measurement

    • The balanced scorecard is ultimately about choosing measures and targets.
    • Its various design methods are meant to help identify these measures and targets, usually by a process of abstraction that narrows the search space for a measure.
    • For instance, a company can find a measure to inform about a particular objective within the Customer perspective rather than find a measure for customers in general.
    • Useful measurement feedback from a balanced scorecard is also essential.
    • This means that careful consideration is required when interpreting applicable measurements.

  • Accuracy, Precision, and Error

    • Accuracy is how close a measurement is to the correct value for that measurement.
    • The precision of a measurement system is refers to how close the agreement is between repeated measurements (which are repeated under the same conditions).
    • All measurements would therefore be overestimated by 0.5 g.
    • Unless you account for this in your measurement, your measurement will contain some error.
    • The mean deviates from the "true value" less as the number of measurements increases.