Definitions, theorems, and examples related to measurable functions in real analysis.
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The image shows definitions, theorems, and examples related to measurable functions in real analysis, specifically focusing on properties like measurability preservation under certain conditions and the concept of essential supremum.
Answer
Measurable functions: theorems, definitions, and examples include if f = g a.e. then g is measurable; if {fi} converges a.e. to f, then f is measurable; f+ = max (f, 0) and f- = - min (f, 0) are measurable; the set of convergence points is measurable; essential supremum definition; and f < ess sup f, a.e.
Here are some definitions, theorems, and examples related to measurable functions:
- Theorem 15: If f is measurable and f = g a.e., then g is measurable.
- Example 13: If {fi} is a sequence of measurable functions converging a.e. to f, then f is measurable, since f = lim sup fi, a.e.
- Example 14: If f is a measurable function, then so are f+ = max (f, 0) and f- = - min (f, 0).
- Example 15: The set of points on which a sequence of measurable functions {fn} converges is measurable.
- Definition 10: Let f be a measurable function; then inf[α: f < α a.e.] is called the essential supremum of f, denoted by ess sup f.
- Example 16: f < ess sup f, a.e.
Answer for screen readers
Here are some definitions, theorems, and examples related to measurable functions:
- Theorem 15: If f is measurable and f = g a.e., then g is measurable.
- Example 13: If {fi} is a sequence of measurable functions converging a.e. to f, then f is measurable, since f = lim sup fi, a.e.
- Example 14: If f is a measurable function, then so are f+ = max (f, 0) and f- = - min (f, 0).
- Example 15: The set of points on which a sequence of measurable functions {fn} converges is measurable.
- Definition 10: Let f be a measurable function; then inf[α: f < α a.e.] is called the essential supremum of f, denoted by ess sup f.
- Example 16: f < ess sup f, a.e.
More Information
Measurable functions are fundamental in real analysis, especially in the context of Lebesgue integration. They generalize the concept of continuous functions and allow for the integration of a broader class of functions.
Tips
Pay close attention to the definitions and theorems related to measurable functions, as they form the basis for many advanced concepts in real analysis and probability theory.
Sources
- Measurable Functions - Definition, Properties and Examples - BYJU'S - byjus.com
- Measurable function - Wikipedia - en.wikipedia.org
- [PDF] Chapter 3. Measurable Functions - UC Davis Math - math.ucdavis.edu
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