One of the most common misconceptions when first learning measure theory is the belief that Lebesgue integration was invented because the Riemann integral could not handle discontinuous functions.

This is not quite true.

In fact, the Riemann integral can handle many discontinuities perfectly well. The real limitation lies elsewhere.

Understanding this distinction helps explain why Lebesgue’s ideas were revolutionary.

The Common Misconception

Many students encounter examples involving functions that are discontinuous at a few points and are told that Lebesgue integration is more powerful.

This can create the impression that:

Riemann integration fails whenever a function has discontinuities.

This is false.

Consider the function:

$$f(x)=\mathbf{1}_{{0}}(x)$$

defined on the interval ([0,1]).

Explicitly,

$$f(x)=\begin{cases}1,&x=0\0,&x\neq0\end{cases}$$

This function is discontinuous at (x=0).

Nevertheless, it is Riemann integrable and:

$$\int_0^1 f(x),dx=0$$

Why?

Because changing the value of a function at a single point contributes no area.

Thus isolated discontinuities are not a problem for Riemann integration.

The Real Criterion for Riemann Integrability

The key result is the following theorem.

Lebesgue’s Criterion for Riemann Integrability

Let (f:[a,b]\to\mathbb{R}) be a bounded function. Then (f) is Riemann integrable if and only if the set of points where (f) is discontinuous has Lebesgue measure zero.

Symbolically:

$$f\text{ is Riemann integrable}\iff m(D_f)=0$$

where (D_f) denotes the set of discontinuities of (f).

This theorem completely characterizes which bounded functions are Riemann integrable.

The issue is therefore not whether discontinuities exist.

The issue is how large the set of discontinuities is.

A Function That Breaks Riemann Integration

Consider the Dirichlet function:

$$f(x)=\begin{cases}1,&x\in\mathbb{Q}\0,&x\notin\mathbb{Q}\end{cases}$$

This function equals:

• 1 on rational numbers,
• 0 on irrational numbers.

Every interval contains both rational and irrational numbers.

Consequently, the function is discontinuous at every point.

The set of discontinuities is:

$$D_f=[0,1]$$

and therefore:

$$m(D_f)=1$$

which is certainly not zero.

By Lebesgue’s Criterion, the function is not Riemann integrable.

Lebesgue’s Perspective

Lebesgue looked at the same function and asked a different question.

Instead of focusing on discontinuities, he focused on the size of the sets where the function takes particular values.

The rational numbers satisfy:

$$m(\mathbb{Q}\cap[0,1])=0$$

while the irrational numbers satisfy:

$$m([0,1]\setminus\mathbb{Q})=1$$

The function is equal to 1 only on a set of measure zero.

Therefore:

$$\int_0^1 f(x),dx=0$$

in the Lebesgue sense.

What Riemann could not integrate, Lebesgue integrates effortlessly.

The Philosophical Shift

Riemann’s approach is geometric.

It asks:

How does the function behave as we partition the x-axis?

Lebesgue’s approach is measure-theoretic.

It asks:

How large are the sets on which the function takes various values?

This shift from intervals to measurable sets is the fundamental innovation of Lebesgue’s theory.

Why This Matters

Many important functions arising in:

• Probability theory,
• Functional analysis,
• Harmonic analysis,
• Ergodic theory,
• Partial differential equations,

have complicated discontinuity structures.

Riemann integration is often too restrictive.

Lebesgue integration succeeds because it cares about the measure of exceptional sets rather than their mere existence.

The Deep Insight

The true limitation of Riemann integration is not isolated discontinuities.

It is the inability to cope with functions whose bad behavior occurs on sets that are too large.

Lebesgue’s great insight was to replace the question:

Where are the discontinuities?

with the question:

How large is the set where the bad behavior occurs?

That simple change transformed the theory of integration and became one of the foundations of modern analysis.

References

1. Tao, T. (2011). An Introduction to Measure Theory. American Mathematical Society.
2. Royden, H. L., & Fitzpatrick, P. M. (2010). Real Analysis (4th ed.). Pearson.
3. Rudin, W. (1987). Real and Complex Analysis (3rd ed.). McGraw-Hill.
4. Folland, G. B. (1999). Real Analysis: Modern Techniques and Their Applications (2nd ed.). Wiley.
5. Billingsley, P. (1995). Probability and Measure (3rd ed.). Wiley.
6. Cohn, D. L. (2013). Measure Theory (2nd ed.). Birkhäuser.