To integrate a complicated nonnegative function, first replace its values by finitely many levels. For example, round values down to multiples of one-quarter and cap them at two. The result is a simple function: it takes finitely many values on measurable level sets. Refining the grid and raising the cap produces an increasing sequence that approaches the original function at every point.
Predict
Predict why rounding down is useful. It guarantees the approximation never exceeds the target and supports monotone convergence later. A nearest-value approximation may be accurate but can move up and down as the grid changes. The cap is equally important: a dyadic grid alone still has infinitely many possible values when the function is unbounded, so it would not be simple.
Experiment
For the fixed value f=7/10, inspect successive lower approximations before the cap matters.
n
Grid spacing
Lower value
Error
1
1/2
1/2
1/5
2
1/4
1/2
1/5
3
1/8
5/8
3/40
A refinement need not strictly improve every point, but it cannot lower the approximant. Compute the next row and compare the error with the new spacing.
Understand
A function f from (X, F) to the extended real line is measurable when preimages of Borel sets are in F. It is enough to test threshold sets where f>t for real t, because these rays generate the Borel sigma-algebra and preimages preserve sigma-algebra operations. Countable suprema are measurable since sup f_n>t holds exactly when some f_n>t. Infima follow by complements or negation, and liminf and limsup follow by countable combinations. For real-valued functions, sums and products are measurable by composing the measurable pair map with continuous arithmetic; avoid undefined expressions such as infinity minus infinity when using extended values.
Apply
For nonnegative f define s_n=min (n,2^(−n)floor (2^n f)), with value n where f is infinite. Each s_n has finite range, and its level sets are measurable intersections of threshold sets. Refining a dyadic grid never lowers its rounded-down value; increasing the cap also never lowers it. Thus s_n increases. If f(x) is finite, eventually the cap exceeds it and the error is below 2^(−n); if f(x)=∞, the approximants equal n and diverge upward. Hence s_n increases pointwise to f. This is a constructive theorem, not an assumption that measurable functions already look like step functions on intervals.
Continue
The level sets of a simple function need not be intervals or geometrically simple. Simple refers to finitely many values, not to an easy drawing of its domain. On a finite measurable space, a measurable function must be constant on every atom; otherwise a threshold would distinguish two points the sigma-algebra cannot separate. For functions equal almost everywhere, measurability of a modified version is automatic on a complete space but may fail on an uncompleted one if arbitrary null subsets are used. These distinctions will matter when defining integrals of equivalence classes.
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For f(x)=7/10, n=2, compute 2^(−n)floor (2^n f(x)).
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
For f=3 times the indicator of A, how many distinct values does f take if A and its complement are nonempty?
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
Why do measurable preimages of rays (t,∞) suffice for real-valued measurability?
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Prove a pointwise supremum of countably many measurable functions is measurable.
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Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Explain why dyadic lower approximants with cap n increase to a nonnegative measurable f.
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Prove a continuous real function composed with a measurable real function is measurable.
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Reflect and continue
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