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Swapping the Order of Supremum Operators: Conditions and Implications

March 16, 2025Technology2422
Swapping the Order of Supremum Operators: Conditions and Implications

Swapping the Order of Supremum Operators: Conditions and Implications

The question of whether the order of the supremum operators can be swapped arises frequently in mathematical analysis, particularly in optimization and functional analysis. This article delves into the conditions under which supx supy fxy is equal to supy supx fxy. We explore various scenarios, providing examples and theoretical insights.

General Case

It is not always the case that a b, where a supx supy fxy and b supy supx fxy. The equality depends on the properties of the function fxy and the sets over which x and y are defined.

Example

Consider the function fxy -xy for x, y u2208 [0, 1].

To calculate a, we start with:

a supx supy fxy supx supy -xy supx -x u2217 1 supx -x 0

To calculate b, we have:

b supy supx fxy supy supx -xy supy -1 u2217 y supy -y 0

In this case, we find a b 0. However, this does not guarantee that they will always be equal.

Conditions for Equality

Continuity and Compact Sets

If fxy is continuous and the sets of x and y are compact, then a b. This is a consequence of the properties of continuous functions on compact sets, which ensure that the supremum is attained.

Monotonicity

If fxy is monotonic in both variables, i.e., increasing x or y does not decrease f, then a b.

If fxy is measurable and the order of integration can be interchanged, the Fubini-Tonelli theorem applies, allowing for the swapping of the order of the supremum.

Uniformity

If the function fxy does not vary significantly between the two dimensions or can be expressed in a separable form, such as fxy gx hy, then a b.

Proving the Equality

To prove that a b, we can prove both directions of a non-strict inequality. Define:

gy supx fxy

hx supy fxy

By the definition of supremum, for all x, y:

gy u2264 b hx u2264 a

Further, since fxy u2264 gy and fxy u2264 hx for all x, y, taking the supremum yields:

fxy u2264 hx u2192 gy u2264 a u2192 b u2264 a fxy u2264 gy u2192 hx u2264 b u2192 a u2264 b

It is clear that the key operation is the ability to maintain non-strict inequalities while taking supremums. This is a straightforward application of the definition of supremum. Assuming iz u2264 jz for some functions iz and jz and for all z, we have:

supz iz u2264 supz jz

Assuming the contrary, supz iz > supz jz, leads to a contradiction as shown above.

Conclusion

In summary, the equality a b can be guaranteed under certain conditions such as continuity and compactness, monotonicity, measurability, and uniformity. However, it is essential to analyze the specific properties of the function fxy and the domains of x and y to determine whether the order of supremum can be swapped.

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